calculus/1 Differential and Derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differentiation\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that $f(x)$ is continuous on $[a, b]$ and let $P = (x_0, f (x_0))$\n",
    "and $Q = (x_0 + h, f (x_0 + h))$ on the graph of $f (x)$:\n",
    "<img src=\"imgs/1/ch3-1.png\" width=60% />\n",
    "Then the slope of $\\overset{\\longleftrightarrow}{P Q}$, secant line passing\n",
    "through $P$ and $Q$, is\n",
    "\n",
    "$$\n",
    "\\mathbf{m = \\lim\\limits_{h \\rightarrow 0} \\frac{f (x_0 + h) - f (x_0)}{h} }\n",
    "$$\n",
    "\n",
    "so called Newton quotient of $f (x)$ at $x = x_0$. If the limit of $m$ as $h$\n",
    "approaches $0$ exists, it is called the derivative of $f (x)$ at $x = x_0$,\n",
    "i.e. it is the slope of tangent line of $f (x)$ passing through $(x_0, f\n",
    "(x_0))$, and denoted as $f' (x_0)$. The process for finding derivative is\n",
    "called differentiation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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7Crlo7Zw5c1i8eDGqysknn5y7/auvvqJp06bUqVOH6tWr06NHD2ZEuEDQokWL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\ntQ9aW4VWAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10db8b978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(-2,2,101)\n",
    "plt.plot(x,x**2,\"-b\")\n",
    "plt.plot([0,2.5],[-1,4],'-r')\n",
    "plt.scatter([1],[1])\n",
    "plt.text(1.5,1,'$(x_0,f(x_0))$',size=14)\n",
    "plt.text(2.5,2,'slope=$f\\'(x_0)$')\n",
    "plt.title('tangent $y=f(x_0)+f\\'(x_0)(x-x_0)$',size=14)\n",
    "plt.axis(\"equal\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remarks\n",
    "---\n",
    "There are several symbols for derivatives as follows:\n",
    "1. the derivatives value of $f (x)$ at $x = x_0$:\n",
    "    \n",
    "$$ f' (x_0) = \\left.\\frac{d}{d x} f (x) \\right|_{x = x_0} = \\frac{d}{d x} f (x_0) =\n",
    "D_x f (x_0) $$\n",
    "\n",
    "- the derivative of $f (x)$:\n",
    "    $$ f' (x_0) = \\frac{d}{d x} f (x) = D_x f (x) $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "a). $|x|$ is differentiable everywhere except at $x=0$. <br>\n",
    "b). Neither $x^{1/3}$ nor $x^{2/3}$ are not differentiable at $x=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Theorem\n",
    "---\n",
    "If $f(x)$ is differentiable at $x=a$, then $f(x)$ is continuous at $x=0$.\n",
    "\n",
    "Proof\n",
    "---\n",
    "\\begin{eqnarray}\n",
    "  \\lim\\limits_{x\\to a}|f(x)-f(a)|&=& \\lim\\limits_{x\\to a}\\left|\\frac{f(x)-f(a)}{x-a}\\right|\\cdot |x-a|\\\\\n",
    "  &=& |f'(a)|\\cdot0\\\\\n",
    "  &=& 0\n",
    "\\end{eqnarray}\n",
    "This means $\\lim\\limits_{x\\to a}f(x)=f(a)$, i.e. $f(x)$ is continuous at $x=a$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Suppose that\n",
    "$$ f (x) = \\left\\{ \\begin{array}{l}\n",
    "       x \\sin (1 / x) \\text{, if } x \\neq 0,\\\\\n",
    "       0, \\text{, if } x = 0\n",
    "     \\end{array} \\right. $$\n",
    "  Then $f (x)$ is continuous for any real number by the squeeze value theorem.\n",
    "  And the the derivative at $x = 0$, is as follows:\n",
    "  \\begin{eqnarray*}\n",
    "    f' (0) & = & \\lim_{h \\rightarrow 0} \\frac{f (h) - f (0)}{h}\\\\\n",
    "    & = & \\lim_{h \\rightarrow 0} \\frac{h \\sin (1 / h)}{h}\\\\\n",
    "    & = & \\lim_{h \\rightarrow 0} \\sin (1 / h)\n",
    "  \\end{eqnarray*}\n",
    "  This concludes that the limit fails to exist.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "The derivative of sine function is derived as follows:\n",
    "\\begin{eqnarray*}\n",
    " (\\sin x)' & = & \\lim_{h \\rightarrow 0} \\frac{\\sin (x + h) - \\sin x}{h}\\\\\n",
    " & = & \\lim_{h \\rightarrow 0} \\frac{\\sin x \\cos h + \\cos x \\sin h - \\sin x}{h}\\\\\n",
    " & = & \\lim_{h \\rightarrow 0} \\left( \\sin x \\frac{\\cos h - 1}{h} + \\cos x \\frac{\\sin h}{h} \\right)\\\\\n",
    " & = & \\cos x\n",
    "\\end{eqnarray*}\n",
    "\n",
    "since $\\lim\\limits_{h \\rightarrow 0} \\frac{\\sin h}{h} = 1$ and\n",
    "  $\\lim\\limits_{h \\rightarrow 0} \\frac{\\cos h - 1}{h}$=0.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercises, p153\n",
    "---\n",
    "\n",
    "(61). Let \n",
    "$$ f (x) = \\left\\{ \\begin{array}{l}\n",
    "   x^2 \\sin (1 / x) \\text{, if } x \\neq 0,\\\\\n",
    "   0, \\text{, if } x = 0,\n",
    "\\end{array} \\right. \n",
    "$$\n",
    "&nbsp;&nbsp;  a). $f(x)$ is differentiable at $x=0$ and $f'(0)=0$.<br>\n",
    "&nbsp;&nbsp;   b). Its image for $(x,y)\\in[-0.5,0.5]\\times[-0.1,0.1]$.\n",
    "\n",
    "**True or False**\n",
    "\n",
    "(66). If $f(x)$ is differentiable at $x=a$, and  $g(x)$ is not differentiable at $x=a$, then  $fg(x)$ is not differentiable at $x=a$. (**<code style=\"color:white\">False</code>**). <br>\n",
    "(67) If both $f(x)$  and  $g(x)$ are not differentiable at $x=a$, then  $fg(x)$ is not differentiable at $x=a$. (**<code style=\"color:white\">False</code>**). <br>\n",
    "(68) If both $f(x)$  and  $g(x)$ are not differentiable at $x=a$, then  $f+g$ is not differentiable at $x=a$. (**<code style=\"color:white\">False</code>**). <br>\n",
    "(69). The domain of $f'(x)$ is the same as $f(x)$.  (**<code style=\"color:white\">False</code>**). <br>\n",
    "(70). If $n\\in\\mathbb{N}$, the there exists a function $f$ such that $f$ is differentiable everywhere except at $n$ number. (**<code style=\"color:white\">True </code>**)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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PrQ+EHOVVLaxrCPL9AVkJ/9QTT8457lxZx0NrG/hO7wyuG5XHuJ2ObFlTF+Qni2u4bnEN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\nIh5S+IuIeEjhLyLiof8HKfSR2moVndYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x106f59b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy import sin\n",
    "eps=1e-6\n",
    "x=np.linspace(-0.5,0.5,101)\n",
    "\n",
    "plt.figure(figsize=(6,6))\n",
    "plt.plot(x,x*x*sin(1/(x+eps)))\n",
    "plt.ylim(-0.5,0.5)\n",
    "plt.xlim(-0.5,0.5)\n",
    "plt.fill_between([-0.5,0.5,0.5,-0.5,-0.5],[-0.1,-0.1,0.1,0.1,-0.1],color=\"pink\")\n",
    "plt.axis(\"equal\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2.2 Rules of Differentiation\n",
    "---\n",
    "1. $(c)' = 0$ for any constant $c$.\n",
    "- $(x^n)' = n x^{n - 1}$ for $x > 0$ and $n \\in \\mathbb{R}$. Here, a\n",
    "    example for $n = - 1$:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    " \\left(\\frac{1}{x}\\right)' & = & \\lim_{h \\rightarrow 0} \\frac{\\frac{1}{x + h} -\n",
    " \\frac{1}{x}}{h}\\\\\n",
    " & = & \\lim_{h \\rightarrow 0} \\frac{x - (x + h)}{x h (x + h)}\\\\\n",
    " & = & \\frac{- 1}{x^2}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "- $(f \\pm g)' = f' \\pm g'$ ,    \n",
    "- $(f \\cdot g)' = f' g + g' f$,\n",
    "- $(f / g)' = (f' g - g' f) / g^2$ suppose that $g (x) \\neq 0$ for any\n",
    "    $x$,\n",
    "- (Power Rule) $(x^r)' = r x^{r - 1}$ for $r > 0$,\n",
    "- (Chain's Rule) $(f \\circ g)' = (f (g (x)))' = f' (g (x)) g' (x)$  \n",
    "$$\n",
    " (f (g (x)))' = \\frac{d f (g (x))}{\\color{red}{d\\not{g (x)}}} \\cdot\n",
    "     \\frac{{\\color{red}{d\\not{g (x)}}}}{d x}\n",
    "$$     \n",
    " The following diagram says everything about chain's rule:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "      \\begin{array}{ccccc}\n",
    "        &  &  &  & \\\\\n",
    "        & g(x) &  & f(u) & \\\\\n",
    "        x & \\longrightarrow & u=g(x) & \\longrightarrow & f(g(x))\\\\\n",
    "        & \\downarrow &  & \\downarrow & \\\\\n",
    "        & \\color{brown}{g'(x)} & \\times & \\color{brown}{f'(u)} & \\\\\n",
    "        &  & \\downarrow &  & \\\\\n",
    "        &  & \\color{brown}{f'(g(x))g'(x) }&  & \n",
    "      \\end{array} &  & \n",
    "\\end{eqnarray*}\n",
    "\n",
    "- $(\\sin x)' = \\cos x$, $(\\cos x)' = - \\sin x$, $(\\tan x)' = \\sec^2 x$, $(\\cot x)' = - \\csc^2 x$, $(\\sec x)' = \\sec x \\tan x$, $(\\csc x)' = - \\csc x \\cot x$;\n",
    "- (Logarithmic Rule):\n",
    "   - $(\\ln x)' = (\\log_e x)' = 1 / x$, where $\\ln (\\bullet)$ means\n",
    "      logarithm with base $e \\cong 2.71428 \\cdots$ where $e$, Euler number, is an irrational\n",
    "      number;\n",
    "   - $(\\log_a x)' = 1 / (x \\ln a)$ where $a > 0$;\n",
    "- (Exponential Rules):\n",
    "    - $(e^x)' = e^x$;\n",
    "    - $(a^x)' = a^x \\ln a$, where $a > 0$ and $a \\neq 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\left(\\frac{\\sin x}{1-\\cos x}\\right)'&=&\\frac{(\\sin x)'(1-\\cos x)-\\sin x (1-\\cos x)'}{(1-\\cos x)^2}\\\\\n",
    "  &=& \\frac{\\cos x(1-\\cos x)-\\sin x \\sin x}{(1-\\cos x)^2}\\\\\n",
    "  &=& \\frac{1}{\\cos x -1}\n",
    "\\end{eqnarray}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "$\\left[(1 + x^2)^{100} \\right]' = 2 x \\cdot 100 (1 + x^2)^{99} = 200 x (1 + x^2)^{99}$\n",
    "    \n",
    "\\begin{eqnarray*}\n",
    " \\begin{array}{ccccc}\n",
    "  &  &  &  & \\\\\n",
    "  & 1 + x^2 &  & (\\cdot)^{100} & \\\\\n",
    "  x & \\longrightarrow & 1 + x^2 & \\longrightarrow & (1 + x^2)^{100}\\\\\n",
    "  & \\downarrow &  & \\downarrow & \\\\\n",
    "  & 2 x & \\cdot & 100 (\\cdot)^{99} & \\\\\n",
    "  &  & \\downarrow &  & \\\\\n",
    "  &  & 2 x \\cdot 100 (1 + x^2)^{99} &  & \n",
    " \\end{array} &  & \n",
    "\\end{eqnarray*}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "$\\left(\\sin (x^2)\\right)' = 2 x \\cos x^2$\n",
    "    \n",
    "\\begin{array}{ccccc}\n",
    "  &  &  &  & \\\\\n",
    "  & x^2 &  & \\sin (\\cdot) & \\\\\n",
    "  x & \\longrightarrow & x^2 & \\longrightarrow & \\sin (x^2)\\\\\n",
    "  & \\downarrow &  & \\downarrow & \\\\\n",
    "  & 2 x & \\cdot & \\cos (\\cdot) & \\\\\n",
    "  &  & \\downarrow &  & \\\\\n",
    "  &  & 2 x \\cdot \\cos (x^2)^{} &  & \n",
    "\\end{array}\n",
    "    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "$\\left(\\sin^2 x\\right)' = 2 \\sin x \\cos x$\n",
    "\n",
    "\\begin{array}{ccccc}\n",
    " &  &  &  & \\\\\n",
    " & \\sin x &  & (\\cdot)^2 & \\\\\n",
    " x & \\longrightarrow & \\sin x & \\longrightarrow & \\sin^2 (x)\\\\\n",
    " & \\downarrow &  & \\downarrow & \\\\\n",
    " & \\cos x & \\cdot & 2 (\\cdot) & \\\\\n",
    " &  & \\downarrow &  & \\\\\n",
    " &  & \\cos x \\cdot 2 (\\sin x)^{} &  & \n",
    "\\end{array}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import Symbol, diff,sin,cos,sqrt,pprint,Function,tan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=Symbol(\"x\")\n",
    "u=Symbol(\"u\")\n",
    "def ChainRule(f,g):\n",
    "    u=Symbol(\"u\")\n",
    "    l1=diff(f,u)\n",
    "    l2=diff(g,x)\n",
    "    print(\"f'(g(x)) =\",l1*l2,\", where g(x) = u = \",g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Examples\n",
    "---\n",
    "Differentiate a) $\\sqrt{2x^2-1}$ b). $u=x^3+1, u^3-u^2+u+1$ c). $3\\cos x^2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = 2*x/sqrt(u) , where g(x) = u =  2*x**2 - 1\n"
     ]
    }
   ],
   "source": [
    "ChainRule(sqrt(u),2*x*x-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = 3*x**2*(3*u**2 - 2*u + 1) , where g(x) = u =  x**3 + 1\n"
     ]
    }
   ],
   "source": [
    "ChainRule(u*u*u-u*u+u+1,x*x*x+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = -6*x*sin(u) , where g(x) = u =  x**2\n"
     ]
    }
   ],
   "source": [
    "ChainRule(3*cos(u),x*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = 6*x*(3*tan(u)**2 + 3)*tan(u)**2 , where g(x) = u =  3*x**2 + 1\n"
     ]
    }
   ],
   "source": [
    "ChainRule(tan(u)*tan(u)*tan(u),3*x*x+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Find the tangent line of $f(x)=x^2\\sin 3x$ at $x=\\pi/2$\n",
    "\n",
    "a). $\\left.(x^2\\sin 3x)'=2x\\sin(3x)+x^2\\cdot 3\\cos 3x\\right|_{\\quad x=\\pi/2}=-\\pi$<br>\n",
    "b). $f(\\pi/2)=-\\pi^2/4$<br>\n",
    "c). tangent line: $\\left(y-f(\\pi/2)\\right)=f'(\\pi/2)(x-\\pi/2)$, i.e. $y+\\pi^2/4=-\\pi(x-\\pi/2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x10bbc6ef0>"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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EoaCUKIiIxKiJE2HVKg07RFOnTpqnUFBKFEREYtTIkbas7/DDw44kcXTqBBs2qJ5CQShR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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10addc3c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy import pi,sin\n",
    "x=np.linspace(0,pi,101)\n",
    "plt.plot(x,x*x*sin(3*x))\n",
    "t=x[20:-20]\n",
    "f=-pi*t+pi*pi/4\n",
    "plt.plot(t,f,color='red')\n",
    "plt.scatter([pi/2],[-pi*pi/4])\n",
    "plt.text(1.1,-1,'$(\\pi/2,-\\pi^2/4)$',size=14)\n",
    "plt.text(1,-5,'$y=-\\pi x+\\pi^2/4$',size=14,color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Differentiate the following function, $x^x$.\n",
    "\n",
    "<font color=\"red\">**Right or Wrong**</font>\n",
    "- If it is a power function, then $ (x^x)' = x x^{x - 1} = x^x$;\n",
    "- If it is an exponential function, then $(x^x)' = x^x \\ln x$.\n",
    "  \n",
    "But it is neither power function nor exponential function. Then we have to\n",
    "  modify it to be one of them. As well known, $2 = \\begin{array}{c}\n",
    "    10\n",
    "  \\end{array}^{\\log_{10} 2}$, then\n",
    "\n",
    "\\begin{eqnarray*}\n",
    " & x^x = e^{x \\log_e x} & = e^{x \\ln x}\n",
    "\\end{eqnarray*}\n",
    "  \n",
    "Then\n",
    "  \n",
    "\\begin{array}{ccccc}\n",
    " &  &  &  & \\\\\n",
    " & x \\ln x &  & e^{(\\cdot)} & \\\\\n",
    " x & \\longrightarrow & x \\ln x & \\longrightarrow & e^{x \\ln x}\\\\\n",
    " & \\downarrow &  & \\downarrow & \\\\\n",
    " & \\ln x + 1 & \\cdot & e^{(\\cdot)} & \\\\\n",
    " &  & (\\ln x + 1) \\cdot e^{x \\ln x} &  & \n",
    "\\end{array}\n",
    "  \n",
    "  i.e. $(x^x)'=(1 + \\ln x) x^x$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Higher-order Derivatives\n",
    "---\n",
    "Suppose that $f(x)$ is smooth function; the second order derivative is defined as follows:\n",
    "$$ f'' (x) = \\frac{d^2}{d x^2} f (x)  = D_x^2 f (x) =\\frac{d}{d x} f'(x) $$\n",
    "And the more higher order derivatives, ($n$-th)-order for instance, are defined by the recurive formula:\n",
    "\\begin{eqnarray}\n",
    " f^{(3)}(x)&=& \\frac{d^3}{d x^3} f (x)  = D_x^3 f (x)=  \\frac{d}{d x} f'' (x) \\\\\n",
    " &\\vdots&\\\\\n",
    " f^{(n)} (x) &=& \\frac{d^n}{d x^n} f (x)  = D_x^n f (x)=  \\frac{d}{d x} f^{(n-1)} (x) \n",
    "\\end{eqnarray} \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Investigate the derivatives of $f (x) = x^2 \\sin x$.\n",
    "  \n",
    "**Sol:**\n",
    "  \n",
    " The first order of derivative is:\n",
    "\\begin{eqnarray*}\n",
    " f' (x) & = & (x^2 \\sin x)'\\\\\n",
    " & = & (x^2)' \\sin x + x^2 (\\sin x)'\\\\\n",
    " & = & 2 x \\sin x + x^2 \\cos x\n",
    "\\end{eqnarray*}\n",
    "And the second order derivative is:\n",
    "\\begin{eqnarray*}\n",
    "  f'' (x) & = & (x^2 \\sin x)'' \\\\\n",
    "  & = & (2 x \\sin x + x^2 \\cos x)'\\\\\n",
    "  & = & (2 x)' \\sin x + 2 x (\\sin x)' + (x^2)' \\cos x + x^2 (\\cos x)'\\\\\n",
    "  & = & 2 \\sin x + 2 x \\cos x + 2 x \\cos x - x^2 \\sin x\\\\\n",
    "  & = & 2 \\sin x + 4 x \\cos x - x^2 \\sin x\n",
    "\\end{eqnarray*}\n",
    "Exactly, the last result is equal to the following:\n",
    "$$\n",
    "C (2, 0) (x^2)'' \\sin x + C (2, 1) (x^2)' (\\sin x)' + C (2, 2) x^2 (\\sin\n",
    "     x)'' \n",
    "$$\n",
    "where $C (n, k) = n! / k! (n - k)$! and $n! = 1 \\cdot 2 \\cdot 3\n",
    "  \\cdots n$, the factorial of $n$. \n",
    "  \n",
    "This relation can be extended to the more higher differentiation:\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "  f^{(n)} & = & (x^2 \\sin x)^{(n)}\\\\\n",
    "  & = & C (n, 2) (x^2)'' (\\sin x)^{(n - 2)} + C (n, 1) (x^2)' (\\sin x)^{(n- 1)} + C (n, 0) x^2 (\\sin x)^{(n)}\\\\\n",
    "  & = & n (n - 1) (\\sin x)^{(n - 2)} + 2 n x (\\sin x)^{(n - 1)} + x^2 (\\sin x)^{(n)}\n",
    "\\end{eqnarray*}\n",
    "\n",
    "By the last result, the derivative of $10$-th order is\n",
    "$$ f^{(10)} (x) = 90 \\sin x + 20 x \\cos x - x^2 \\sin x $$\n",
    "  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differention with Sympy\n",
    "---"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Symbol, diff,sin,cos,exp,pi,pprint,Abs\n",
    "x=Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-x**2*sin(x) + 20*x*cos(x) + 90*sin(x)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x*x*sin(x),x,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1). the 1-order derivative is: \n",
      " 2                    \n",
      "x ⋅cos(x) + 2⋅x⋅sin(x)\n",
      "\n",
      "\n",
      "2). the 2-order derivative is: \n",
      "   2                               \n",
      "- x ⋅sin(x) + 4⋅x⋅cos(x) + 2⋅sin(x)\n",
      "\n",
      "\n",
      "3). the 3-order derivative is: \n",
      "   2                               \n",
      "- x ⋅cos(x) - 6⋅x⋅sin(x) + 6⋅cos(x)\n",
      "\n",
      "\n",
      "4). the 4-order derivative is: \n",
      " 2                                \n",
      "x ⋅sin(x) - 8⋅x⋅cos(x) - 12⋅sin(x)\n",
      "\n",
      "\n",
      "5). the 5-order derivative is: \n",
      " 2                                 \n",
      "x ⋅cos(x) + 10⋅x⋅sin(x) - 20⋅cos(x)\n",
      "\n",
      "\n",
      "6). the 6-order derivative is: \n",
      "   2                                 \n",
      "- x ⋅sin(x) + 12⋅x⋅cos(x) + 30⋅sin(x)\n",
      "\n",
      "\n",
      "7). the 7-order derivative is: \n",
      "   2                                 \n",
      "- x ⋅cos(x) - 14⋅x⋅sin(x) + 42⋅cos(x)\n",
      "\n",
      "\n",
      "8). the 8-order derivative is: \n",
      " 2                                 \n",
      "x ⋅sin(x) - 16⋅x⋅cos(x) - 56⋅sin(x)\n",
      "\n",
      "\n",
      "9). the 9-order derivative is: \n",
      " 2                                 \n",
      "x ⋅cos(x) + 18⋅x⋅sin(x) - 72⋅cos(x)\n",
      "\n",
      "\n",
      "10). the 10-order derivative is: \n",
      "   2                                 \n",
      "- x ⋅sin(x) + 20⋅x⋅cos(x) + 90⋅sin(x)\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "for i in range(1,11):\n",
    "    print(\"%d). the %d-order derivative is: \" %(i,i))\n",
    "    pprint(diff(x*x*sin(x),x,i))\n",
    "    print(\"\\n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "u=Symbol(\"u\")\n",
    "def ChainRule(f,g):\n",
    "    u=Symbol(\"u\")\n",
    "    l1=diff(f,u).subs({u:g})\n",
    "    l2=diff(g,x)\n",
    "    print(\"f'(g(x)) =\",l1*l2,\", where u = \",g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "u=Symbol(\"u\")\n",
    "def ChainRule2(f,g):\n",
    "    u=Symbol(\"u\")\n",
    "    l1=diff(f,u).subs({u:g})\n",
    "    l2=diff(g,x)\n",
    "    print(\"(f(g(x)))' =\",l1*l2,\", \\nwhere f(u) = \",f, \" and u = g = \",g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "u=Symbol(\"u\")\n",
    "def ChainRule3(f,g):\n",
    "    u=Symbol(\"u\")\n",
    "    fu=diff(f,u)\n",
    "    fx=diff(f,u).subs({u:g})\n",
    "    gx=diff(g,x)\n",
    "    fg=f.subs({u:g})\n",
    "    print(\"   \",g,\"\\t\",f,\"\\n\", \\\n",
    "          \"x\",\"  ⟶  \",g,\"   ⟶   \",fg, \"\\n\",\\\n",
    "          \"     ↓     ×       ↓\", \"\\n\",\\\n",
    "          \"   \",gx,\"\\t\",fu,\"\\n\",\\\n",
    "          \"           ↓    \\n\", \\\n",
    "          \"     \", gx*fx\n",
    "         )\n",
    "    #print(\"(f(g(x)))' =\",l1*l2,\", \\nwhere f(u) = \",f, \" and u = g = \",g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "u=Symbol(\"u\")\n",
    "def ChainRule3(f,g):\n",
    "    \"\"\"\n",
    "    Differentiate f(g(x))\n",
    "    f=f(u)\n",
    "    g=u=g(x)\n",
    "    e.g. (sin(x**2+1))' uses\n",
    "    ChainRule3(sin(u),x**2)\n",
    "    \"\"\"\n",
    "    u=Symbol(\"u\")\n",
    "    fu=str(diff(f,u))\n",
    "    fx=str(diff(f,u).subs({u:g}))\n",
    "    gx=str(diff(g,x))\n",
    "    fgx= str(diff(g,x)*diff(f,u).subs({u:g}))\n",
    "    gfunc=str(g)\n",
    "    ffunc0=str(f)\n",
    "    ffunc=str(f.subs({u:g}))\n",
    "    print('{:3}'.format(\" \"),'{:10.8}'.format(gfunc),'{:12}'.format(\" \"),'{:8.8}'.format(ffunc0))\n",
    "    print('{:3}'.format(\"x\"),'{:4}'.format(\"⟶\"),'u =','{:12}'.format(gfunc),\n",
    "          '{:5}'.format(\"⟶\"),'{:12}'.format(ffunc))\n",
    "    print('{:3}'.format(\" \"),'{:20}'.format(\"↓\"),'{:2}'.format(\" \"),\"↓\")\n",
    "    print('{:3}'.format(\" \"),'{:10.8}'.format(gx),'{:8}'.format(\" × \"),'{:2}'.format(\" \"),'{:8}'.format(fu))\n",
    "    print('{:15}'.format(\" \"),\"↓\")\n",
    "    print('(',ffunc,\")'=\",'{:15}'.format(fgx))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "?ChainRule3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = 2*sin(x)*cos(x) , where u =  sin(x)\n"
     ]
    }
   ],
   "source": [
    "ChainRule(u**2,sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(f(g(x)))' = 2*x*cos(x**2) , \n",
      "where f(u) =  sin(u)  and u = g =  x**2\n"
     ]
    }
   ],
   "source": [
    "ChainRule2(sin(u),x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    x**2                    sin(u)  \n",
      "x   ⟶    u = x**2         ⟶     sin(x**2)   \n",
      "    ↓                       ↓\n",
      "    2*x         ×          cos(u)  \n",
      "                ↓\n",
      "( sin(x**2) )'= 2*x*cos(x**2)  \n"
     ]
    }
   ],
   "source": [
    "ChainRule3(sin(u),x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    cos(x)                  u**2    \n",
      "x   ⟶    u = cos(x)       ⟶     cos(x)**2   \n",
      "    ↓                       ↓\n",
      "    -sin(x)     ×          2*u     \n",
      "                ↓\n",
      "( cos(x)**2 )'= -2*sin(x)*cos(x)\n"
     ]
    }
   ],
   "source": [
    "ChainRule3(u**2,cos(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    x**2 + x                sin(u)  \n",
      "x   ⟶    u = x**2 + x     ⟶     sin(x**2 + x)\n",
      "    ↓                       ↓\n",
      "    2*x + 1     ×          cos(u)  \n",
      "                ↓\n",
      "( sin(x**2 + x) )'= (2*x + 1)*cos(x**2 + x)\n"
     ]
    }
   ],
   "source": [
    "ChainRule3(sin(u),x**2+x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    x**2 + x                -cos(u) \n",
      "x   ⟶    u = x**2 + x     ⟶     -cos(x**2 + x)\n",
      "    ↓                       ↓\n",
      "    2*x + 1     ×          sin(u)  \n",
      "                ↓\n",
      "( -cos(x**2 + x) )'= (2*x + 1)*sin(x**2 + x)\n"
     ]
    }
   ],
   "source": [
    "ChainRule3(cos(pi-u),x**2+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise, p170\n",
    "---\n",
    "**16.**\n",
    "$$\\left(\\frac{2t^2-3t^{3/2}}{5t^{1/2}}\\right)'$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "          ⎛   2      1.5⎞                          \n",
      "     -0.5 ⎜2⋅t    3⋅t   ⎟    0.5 ⎛       0.5   4⋅t⎞\n",
      "0.5⋅t    ⋅⎜──── - ──────⎟ + t   ⋅⎜- 0.9⋅t    + ───⎟\n",
      "          ⎝ 5       5   ⎠        ⎝              5 ⎠\n"
     ]
    }
   ],
   "source": [
    "#16\n",
    "t=Symbol(\"t\")\n",
    "pprint(diff((2*t**2-3*t**(3/2))/5*t**(1/2),t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**36.**\n",
    "$$\\left(\\frac{x}{x^4-2x^2-1}\\right)'\\left|_{x=-1}\\right.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  ⎛     3      ⎞                \n",
      "x⋅⎝- 4⋅x  + 4⋅x⎠         1      \n",
      "──────────────── + ─────────────\n",
      "               2    4      2    \n",
      "⎛ 4      2    ⎞    x  - 2⋅x  - 1\n",
      "⎝x  - 2⋅x  - 1⎠                 \n"
     ]
    }
   ],
   "source": [
    "#36\n",
    "df=diff((x)/(x**4-2*x**2-1),x)\n",
    "pprint(df)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1/2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.subs({x:-1})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise, p190\n",
    "---\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(20).\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\left(\\frac{1-\\tan x}{1+\\cot x}\\right)'&=&\\frac{(1-\\tan x)'(1+\\cot x)-(1-\\tan x) (1+\\cot x)'}{(1+\\cot x)^2}\\\\\n",
    "  &=& \\color{white}{\\frac{-\\sec^2 x(1+\\cot x)-(1-\\tan x) (-\\csc^2x)}{(1+\\cot x)^2}}\\\\\n",
    "\\end{eqnarray}\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(48).\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\lim\\limits_{h\\to0}\\frac{\\frac{1}{\\sin(x+h)}-\\frac{1}{\\sin x }}{h}&=&\\lim\\limits_{h\\to0}\\frac{\\csc(x+h)-\\csc x }{h}\\\\\n",
    "  &=& \\color{white}{(\\csc x)'=-\\csc x \\cot x}\\\\\n",
    "\\end{eqnarray}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise, p190\n",
    "---\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**12.**\n",
    "$$\\left(\\frac{\\cos\\theta}{1-\\sin\\theta}\\right)'$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import diff, Symbol,symbols,sin,cos,pprint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,t=symbols(\" x t\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                      2       \n",
      "     sin(x)        cos (x)    \n",
      "- ─────────── + ──────────────\n",
      "  -sin(x) + 1                2\n",
      "                (-sin(x) + 1) \n"
     ]
    }
   ],
   "source": [
    "#12\n",
    "pprint(diff(cos(x)/(1-sin(x)),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**22.**\n",
    "$$\\left(\\frac{a\\sin\\theta}{1+b\\cos\\theta}\\right)'$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "         2                    \n",
      "  a⋅b⋅sin (x)       a⋅cos(x)  \n",
      "─────────────── + ────────────\n",
      "              2   b⋅cos(x) + 1\n",
      "(b⋅cos(x) + 1)                \n"
     ]
    }
   ],
   "source": [
    "#22\n",
    "a,b =symbols(\"a b\")\n",
    "pprint(diff(a*sin(x)/(1+b*cos(x)),x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "**42.** Foe $n=0,1,2,\\cdot$,\n",
    "$$(\\cos\\theta)^{(n)}=?$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sin(x)\n",
      "cos(x)\n",
      "-sin(x)\n",
      "-cos(x)\n",
      "sin(x)\n",
      "cos(x)\n",
      "-sin(x)\n",
      "-cos(x)\n"
     ]
    }
   ],
   "source": [
    "# 42\n",
    "for n in range(8):\n",
    "    pprint(diff(sin(x),x,n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Whatever $\\sin x$ or $\\cos x$, absolute value of its derivative is no more than 1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise, p201\n",
    "---\n",
    "**12.**\n",
    "$$\n",
    "\\left(\\left(\\frac{x^2+3}{x}  \\right)^{-2} \\right)'\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "        3              \n",
      "     4⋅x         2⋅x   \n",
      "- ───────── + ─────────\n",
      "          3           2\n",
      "  ⎛ 2    ⎞    ⎛ 2    ⎞ \n",
      "  ⎝x  + 3⎠    ⎝x  + 3⎠ \n"
     ]
    }
   ],
   "source": [
    "# 12\n",
    "pprint(diff(((x*x+3)/x)**(-2),x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**14.**\n",
    "$$\\left(\n",
    "(2x-1)^2(x^2+1)^3\n",
    "\\right)'$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                       2                     3\n",
      "             2 ⎛ 2    ⎞              ⎛ 2    ⎞ \n",
      "6⋅x⋅(2⋅x - 1) ⋅⎝x  + 1⎠  + (8⋅x - 4)⋅⎝x  + 1⎠ \n"
     ]
    }
   ],
   "source": [
    "pprint(diff((2*x-1)**2*(x**2+1)**3,x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**22.** Evaluate $f'(x)$ where\n",
    "\n",
    "$$\\left(\n",
    "\\frac{x+2}{x-3}\n",
    "\\right)^{3/2}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       1.5                              \n",
      "⎛x + 2⎞            ⎛ 1.5    1.5⋅(x + 2)⎞\n",
      "⎜─────⎟   ⋅(x - 2)⋅⎜───── - ───────────⎟\n",
      "⎝x - 2⎠            ⎜x - 2            2 ⎟\n",
      "                   ⎝          (x - 2)  ⎠\n",
      "────────────────────────────────────────\n",
      "                 x + 2                  \n"
     ]
    }
   ],
   "source": [
    "##22\n",
    "pprint(diff(((x+2)/(x-2))**(3/2),x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**22.** Evaluate $f'(x)$ where\n",
    "\n",
    "$$\\left(\n",
    "\\frac{(t+1)^3}{(t^2+2t)^2}\n",
    "\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                  3             2\n",
      "(-4⋅t - 4)⋅(t + 1)     3⋅(t + 1) \n",
      "─────────────────── + ───────────\n",
      "              3                 2\n",
      "    ⎛ 2      ⎞        ⎛ 2      ⎞ \n",
      "    ⎝t  + 2⋅t⎠        ⎝t  + 2⋅t⎠ \n"
     ]
    }
   ],
   "source": [
    "#24\n",
    "pprint(diff((t+1)**3/(t**2+2*t)**2,t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = 3*u**2*cos(x) , where u =  sin(x)\n"
     ]
    }
   ],
   "source": [
    "#29\n",
    "ChainRule(u**3,sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    sin(x)                  u**3    \n",
      "x   ⟶    u = sin(x)       ⟶     sin(x)**3   \n",
      "    ↓                       ↓\n",
      "    cos(x)      ×          3*u**2  \n",
      "                ↓\n",
      "( sin(x)**3 )'= 3*sin(x)**2*cos(x)\n"
     ]
    }
   ],
   "source": [
    "#29\n",
    "ChainRule3(u**3,sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**32.** $y'$ if $y=\\cos(x^2-3x+1)+\\tan(2/x)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                              ⎛   2⎛2⎞    ⎞\n",
      "                            2⋅⎜tan ⎜─⎟ + 1⎟\n",
      "               ⎛ 2      ⎞     ⎝    ⎝x⎠    ⎠\n",
      "- (2⋅x - 3)⋅sin⎝x  - 3⋅x⎠ - ───────────────\n",
      "                                    2      \n",
      "                                   x       \n"
     ]
    }
   ],
   "source": [
    "from sympy import tan\n",
    "pprint(diff(cos(x*x-3*x)+1+tan(2/x),x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**36.** $z'$ if $z=(1+\\csc^2x)^4$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                3               \n",
      "   ⎛   2       ⎞            2   \n",
      "-8⋅⎝csc (x) + 1⎠ ⋅cot(x)⋅csc (x)\n"
     ]
    }
   ],
   "source": [
    "from sympy import csc\n",
    "z=(1+(csc(x))**2)**4\n",
    "pprint(diff(z,x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**42.** $y'$ if $$y=\\frac{x+\\sin2x}{2+\\cos3x}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3⋅(x + sin(x))⋅sin(3⋅x)    cos(x) + 1 \n",
      "─────────────────────── + ────────────\n",
      "                  2       cos(3⋅x) + 2\n",
      "    (cos(3⋅x) + 2)                    \n"
     ]
    }
   ],
   "source": [
    "from sympy import sin,cos\n",
    "y=(x+sin(x))/(2+cos(3*x))\n",
    "pprint(diff(y,x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**52.** $y'$ if $y=x\\tan^2(2x+3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  ⎛     2             ⎞                   2         \n",
      "x⋅⎝4⋅tan (2⋅x + 3) + 4⎠⋅tan(2⋅x + 3) + tan (2⋅x + 3)\n"
     ]
    }
   ],
   "source": [
    "from sympy import tan\n",
    "y=x*tan(2*x+3)**2\n",
    "pprint(diff(y,x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**60.** $y'$ if $y=x\\sin{1/x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "            ⎛1⎞\n",
      "         cos⎜─⎟\n",
      "   ⎛1⎞      ⎝x⎠\n",
      "sin⎜─⎟ - ──────\n",
      "   ⎝x⎠     x   \n"
     ]
    }
   ],
   "source": [
    "y=x*sin(1/x)\n",
    "pprint(diff(y,x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**64.** Find the tangent line of $h(t)=2\\cos^2\\pi t$ at $t=1/4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The tangent line of f(x) at x = 0.25 is (y-1)=-2*pi(x-0.25)\n"
     ]
    }
   ],
   "source": [
    "from sympy import pi,cos,symbols,diff\n",
    "x=symbols(\"x\")\n",
    "h=2*cos(pi*x)**2\n",
    "t0=1/4\n",
    "df=diff(h,x)\n",
    "y0=h.subs({x:t0})\n",
    "m=df.subs({x:t0})\n",
    "\n",
    "print(\"The tangent line of f(x) at x = %s is (y-%s)=%s(x-%s)\" %(t0,y0,m,t0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f'(g(x)) = -3*x**2*sin(u) , where u =  x**3\n"
     ]
    }
   ],
   "source": [
    "#36\n",
    "ChainRule(cos(u),x**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    x**3                    cos(u)  \n",
      "x   ⟶    u = x**3         ⟶     cos(x**3)   \n",
      "    ↓                       ↓\n",
      "    3*x**2      ×          -sin(u) \n",
      "                ↓\n",
      "( cos(x**3) )'= -3*x**2*sin(x**3)\n"
     ]
    }
   ],
   "source": [
    "#36\n",
    "ChainRule3(cos(u),x**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "            ⎛1⎞\n",
      "         cos⎜─⎟\n",
      "   ⎛1⎞      ⎝x⎠\n",
      "sin⎜─⎟ - ──────\n",
      "   ⎝x⎠     x   \n"
     ]
    }
   ],
   "source": [
    "#44\n",
    "pprint(diff(x*sin(1/x),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "    ⎛1⎞ \n",
      "-sin⎜─⎟ \n",
      "    ⎝x⎠ \n",
      "────────\n",
      "    3   \n",
      "   x    \n"
     ]
    }
   ],
   "source": [
    "#60\n",
    "pprint(diff(x*sin(1/x),x,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(77). Find the derivative of \n",
    "$$\\frac{|x|}{\\sqrt{2-x^2}}$$\n",
    "for $x\\ne0$ and make its picture (like bullet-nose).\n",
    "\n",
    "Since \n",
    "$$ |x|= \\left\\{ \\begin{array}{l}\n",
    "               x,\\text{ if } x>0,\\\\\n",
    "              -x,\\text{ if } x<0\\\\\n",
    "               \\end{array}\n",
    "        \\right. \n",
    "$$\n",
    "\n",
    "- $x>0$\n",
    "$$ f'(x)= \\left(\\frac{x}{\\sqrt{2-x^2}}\\right)'=\\frac{2}{(2-x^2)^{3/2}}$$\n",
    "- $x<0$\n",
    "$$ f'(x)= \\left(\\frac{-x}{\\sqrt{2-x^2}}\\right)'=\\frac{-2}{(2-x^2)^{3/2}}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 10)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1086b9128>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy import abs,sqrt\n",
    "epsilon=1e-6\n",
    "x=np.linspace(-sqrt(2)+epsilon,sqrt(2)-epsilon,101)\n",
    "plt.plot(x,abs(x)/sqrt(2-x*x))\n",
    "plt.ylim(0,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(99).\n",
    "---\n",
    "Suppose that\n",
    "$$ f (x) = \\left\\{ \\begin{array}{ll}\n",
    "  x^2 \\sin (1 / x), &\\text{ if } x \\neq 0,\\\\\n",
    "  0, &\\text{ if } x = 0\n",
    "\\end{array} \\right. $$\n",
    "\n",
    "a). For $x\\ne0$,\n",
    "$$ f'(x)= 2x\\sin(1/x)+x^2\\cdot (-1/x^2)\\cos(1/x)=2x\\sin(1/x)-\\cos(1/x)$$\n",
    "b). If $x=0$, then\n",
    "$$ f'(0)=\\lim\\limits_{x\\to0}\\frac{x^2 \\sin (1 / x)-0}{x}=0$$\n",
    "by squeeze value theorem. Thus\n",
    "$$\n",
    "f '(x) = \\left\\{ \\begin{array}{ll}\n",
    " 2x\\sin(1/x)-\\cos(1/x), &\\text{ if } x \\neq 0,\\\\\n",
    "  0, &\\text{ if } x = 0\n",
    "\\end{array} \\right. \n",
    "$$\n",
    "\n",
    "c). For $x\\ne0$,\n",
    "$$ f''(x)= \\left( 2x\\sin(1/x)-\\cos(1/x) \\right)'=2\\sin(1/x)-\\frac{2\\cos(1/x)}{x}-\\frac{\\sin(1/x)}{x^2}$$\n",
    "d). but $f''(0)$ fails to exist since the limit\n",
    "$$ f''(0)=\\lim\\limits_{x\\to0}\\frac{2x\\sin(1/x)-\\cos(1/x)}{x}$$\n",
    "doesn't exist."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       ⎛1⎞      ⎛1⎞\n",
      "2⋅x⋅sin⎜─⎟ - cos⎜─⎟\n",
      "       ⎝x⎠      ⎝x⎠\n"
     ]
    }
   ],
   "source": [
    "pprint(diff(x**2*sin(1/x),x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                ⎛1⎞      ⎛1⎞\n",
      "           2⋅cos⎜─⎟   sin⎜─⎟\n",
      "     ⎛1⎞        ⎝x⎠      ⎝x⎠\n",
      "2⋅sin⎜─⎟ - ──────── - ──────\n",
      "     ⎝x⎠      x          2  \n",
      "                        x   \n"
     ]
    }
   ],
   "source": [
    "pprint(diff(x**2*sin(1/x),x,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/cch/anaconda/lib/python3.5/site-packages/ipykernel/__main__.py:4: RuntimeWarning: divide by zero encountered in true_divide\n",
      "/Users/cch/anaconda/lib/python3.5/site-packages/ipykernel/__main__.py:4: RuntimeWarning: invalid value encountered in sin\n",
      "/Users/cch/anaconda/lib/python3.5/site-packages/ipykernel/__main__.py:4: RuntimeWarning: invalid value encountered in cos\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(-5, 5)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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FIgM2bgTWrs1XRiFLMaMuYVmOPGQUOjqAd99loEBUjoECkQGrVsl9XjMKJmsQ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\niIiIQv0/gk66PpEr8JUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10bdce668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy import sin,cos\n",
    "epsilon=1e-6\n",
    "x=np.linspace(-1,1,401)\n",
    "plt.plot(x,(2*x*sin(1/x)-cos(1/x))/x)\n",
    "plt.ylim(-5,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(100).\n",
    "$$|u|'=(\\sqrt{u^2})'=2u\\cdot u'\\cdot\\frac{1}{2\\sqrt{u^2}}=\\frac{u'u}{|u|}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implicit Differentiation\n",
    "---\n",
    "\n",
    "Definition\n",
    "---\n",
    "  $F (x, y) = 0$ is called equation of $x$ and $y$ of $y$ is called an\n",
    "  implicit function of $x$.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "The circle, $x^2 + y^2 = r^2$, is a famous known equation. And how do we find\n",
    "the tangent line at the point on the circle? Implicit differentiation says\n",
    "that $y$ can be treated as function of $x$ locally, i.e. $y = y (x)$. Then\n",
    "\n",
    "\\begin{eqnarray*}\n",
    "  & x^2 + (y (x))^2 = r^2 & \\\\\n",
    "  \\Longrightarrow & [x^2 + (y (x))^2]' = [r^2]' & \\text{ by differentiating both sides}\\\\\n",
    "  \\Longrightarrow & 2 x + 2 y (x) y' (x) = 0 & \\\\\n",
    "  \\Longrightarrow & y' (x) = - \\frac{x}{y (x)} & \\text{ if }y (x) \\neq 0\n",
    "\\end{eqnarray*}\n",
    "\n",
    "This means that $y' (x)$ depends on both $x$ and $y$. For example,\n",
    "- At $(x, y) = (1 / \\sqrt{2}, 1 / \\sqrt{2})$, \n",
    "  $y' (x) = - \\sqrt{2} /  \\sqrt{2} = -1$;\n",
    "- at $(x, y) = (- 3 / 5, 4 / 5)$, $ y' (x) = - [(- 3 / 5) / (4\n",
    "  / 5)] =  3 / 4$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Find out $y' (x)$ if $x^3 + y^3 - 3 x y = 0$.\n",
    "  \n",
    "By implicit differention:\n",
    "\\begin{eqnarray*}\n",
    "    & [x^3 + y^3 - 3 x y]' & = 0\\\\\n",
    "    \\Longrightarrow & 3 x^2 + 3 y^2 y' - 3 (y + x y') & = 0\\\\\n",
    "    \\Longrightarrow & y'(3 y^2 - 3 x)  & = 3 (y - x^2)\\\\\n",
    "    \\Longrightarrow & y' & = \\frac{ x^2-y}{x-y^2}\n",
    "  \\end{eqnarray*}\n",
    "  if $y^2 - x \\neq 0$ or $x\\ne y^2$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why no Derivative at $(x,y)=(0,0)$ and $\\left(2^{2/3},2^{1/3}\\right)$\n",
    "---\n",
    "<img src=\"imgs/1/sage0.png\" width=40% />\n",
    "\n",
    "It's very clear:\n",
    "- the curve of $x^3+y^3-3xy=0$ crosses origin, $(0,0)$, twice with different tangent; and this is why its tangent can't exist at origin even that the tangent lines exist.  \n",
    "- The latter case, it is obvious that there exists a vertical tangent line at the case, i.e. $\\left.f'\\right|_{(2^{2/3},2^{1/3})}=\\infty$, does not exist.\n",
    "\n",
    "Note\n",
    "---\n",
    "Above image was created by another CAS, <a href=\"www.sagemath.org\">Sage</a>, which owns powerful plotting functionalities for math works. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Function,solve,Derivative\n",
    "y=Function(\"y\")\n",
    "y=y(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3*x**2 - 3*x*Derivative(y(x), x) + 3*y(x)**2*Derivative(y(x), x) - 3*y(x)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l=diff(x**3+y**3-3*x*y,x);\n",
    "l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(x**2 - y(x))/(x - y(x)**2)]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(l,Derivative(y,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ImplicitDiff(express):\n",
    "    l=diff(express,x);\n",
    "    print(\"y'(x) =\",solve(l,Derivative(y,x))[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = (x**2 - y(x))/(x - y(x)**2)\n"
     ]
    }
   ],
   "source": [
    "ImplicitDiff(x**3+y**3-3*x*y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = (-2*x*y(x) + sin(x))/(x**2 + 1)\n"
     ]
    }
   ],
   "source": [
    "ImplicitDiff(x**2*y+y+cos(x)+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = -(3*x**2*y(x)**2 + sin(x))/(2*(x**3 + 2*y(x)**2 - 3*y(x))*y(x))\n"
     ]
    }
   ],
   "source": [
    "ImplicitDiff(y**4-2*y**3+x**3*y**2-cos(x)-8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Evaluate $y'(x)$ for $x\\sin y -y \\cos2x =2x$ at $(x,y)=(\\pi/2,\\pi)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = (-2*y(x)*sin(2*x) - sin(y(x)) + 2)/(x*cos(y(x)) - cos(2*x))\n"
     ]
    }
   ],
   "source": [
    "ImplicitDiff(x*sin(y)-y*cos(2*x)-2*x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus\n",
    "$$ y'(x)\\left |_{(x,y)=(\\pi/2,\\pi)}\\right.=  \\frac{-2\\pi\\sin \\pi-\\sin\\pi+2}{\\pi/2 \\cos \\pi-\\cos\\pi}=\\frac{4}{2-\\pi}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example\n",
    "---\n",
    "Find an equation of tangent line of $4x^4+8x^2y^2-25x^2y+4x^4=0$ at $(x,y)=(2,1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = 2*x*(-8*x**2 - 8*y(x)**2 + 25*y(x))/(16*x**2*y(x) - 25*x**2 + 16*y(x)**3)\n"
     ]
    }
   ],
   "source": [
    "ImplicitDiff(4*x**4+8*x**2*y**2-25*x**2*y+4*y**4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slope is $y'=2\\times2(-8 \\times2^2-8+25)/(16\\times2^2-25\\times2^2+16)=3$. Thus the tangent line is \n",
    "$$ (y-1)=3(x-2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "P. 212\n",
    "\n",
    "#10. Find $y'(x)$ if $\\frac{x+y}{x-y}=y^2+1$, #16. $x+y^2=\\cos xy$\n",
    "<br>\n",
    "#36. tangent line of $2y^2-x^3-x^2=0$ at $(1,1)$.<br>\n",
    "#42. normal line of $x^5-2xy+y^5=0$ at $(1,1)$.<br>\n",
    "#44. tangent line of $x^{2/3}+y^{2/3}=1$ at $(3\\sqrt3,1)$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = (y(x) + 1)/(2*x + y(x) - 1)\n"
     ]
    }
   ],
   "source": [
    "#10.\n",
    "ImplicitDiff((x+y)/(x-y-y**2-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = -(y(x)*sin(x*y(x)) + 1)/(x*sin(x*y(x)) + 2*y(x))\n"
     ]
    }
   ],
   "source": [
    "#16.\n",
    "ImplicitDiff(x+y**2-cos(x*y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = x*(3*x + 2)/(4*y(x))\n"
     ]
    }
   ],
   "source": [
    "#36.\n",
    "ImplicitDiff(2*y**2-x**3-x**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "since $m= 1$, tangent line is $(y-1)=x-1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = (5*x**4 - 2*y(x))/(2*x - 5*y(x)**4)\n"
     ]
    }
   ],
   "source": [
    "#42.\n",
    "ImplicitDiff(x**5-2*x*y+y**5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "slope of normal line is negative inverse of tangent line, $n=-1/m=1$, tangent line is $(y-1)=x-1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y'(x) = -y(x)**(1/3)/x**(1/3)\n"
     ]
    }
   ],
   "source": [
    "#44.\n",
    "ImplicitDiff(x**(2/3)+y**(2/3)-4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "since $m= -\\sqrt3$, tangent line is $(y-1)=-\\sqrt3(x-3\\sqrt3)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Inverse of Function\n",
    "---\n",
    "\n",
    "Inverse Differentiation\n",
    "---\n",
    "Suppose that the inverse of $y=f(x)$ exists and is equal to $x=f^{-1}(y)$. Then\n",
    "$$\\frac{d x}{dy}=\\frac{1}{dy/dx}$$\n",
    "i.e.\n",
    "$$\\left(f^{-1}(y)\\right)'=\\frac{1}{f'(x)}$$\n",
    "\n",
    "Example\n",
    "---\n",
    "The inverse of $\\sin$ is defined as follows:\n",
    "$$ y=\\text{Sin}^{-1} x \\text{ if }x=\\sin(y), \\text{ for } y\\in(-\\pi/2,\\pi/2)$$ \n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\frac{d \\text{Sin}^{-1} x}{d x}&=&\\frac{d y}{d x}\\\\\n",
    "  &=& \\frac{1}{d x/d y}\\\\\n",
    "  &=& \\frac{1}{d(\\sin y)/d y}\\\\\n",
    "  &=& \\frac{1}{\\cos y}=\\frac{1}{\\sqrt{1-\\sin^2y}}=\\frac{1}{\\sqrt{1-x^2}}\\\\\n",
    "\\end{eqnarray}\n",
    "\n",
    "Note\n",
    "---\n",
    "Other derivatives for inverse trigonometric functions are:\n",
    "- $\\left(\\text{Tan}^{-1}(x)\\right)'=\\frac{1}{1+x^2}$,\n",
    "-  $\\left(\\text{Sec}^{-1}(x)\\right)'=\\frac{1}{|x|\\sqrt{x^2-1}}$,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import asin,atan,asec,simplify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/sqrt(-x**2 + 1)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(asin(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  1   \n",
      "──────\n",
      " 2    \n",
      "x  + 1\n"
     ]
    }
   ],
   "source": [
    "pprint(diff(atan(x),x))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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