ML_Calculus.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Machine Learning Calculus"
      ],
      "metadata": {}
    },
    {
      "cell_type": "code",
      "source": [
        "from sympy import *\n",
        "import numpy as np\n",
        "import math\n",
        "\n",
        "x = Symbol('x')\n",
        "y = Symbol('y')"
      ],
      "outputs": [],
      "execution_count": 1,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "limit(1/x**2,x,0)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 2,
          "data": {
            "text/plain": [
              "oo"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 2,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Find the limit of Functions\n",
        "from sympy import Limit, Symbol, S\n",
        "\nLimit(1/x, x, S.Infinity)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 3,
          "data": {
            "text/plain": [
              "Limit(1/x, x, oo, dir='-')"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 3,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# To find the value\n",
        "l = Limit(1/x, x, S.Infinity)\n",
        "l.doit()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 4,
          "data": {
            "text/plain": [
              "0"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 4,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Differentiation\n",
        "diff(15*x**100-3*x**12+5*x-46)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 5,
          "data": {
            "text/plain": [
              "1500*x**99 - 36*x**11 + 5"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 5,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Constant\n",
        "diff(99)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 6,
          "data": {
            "text/plain": [
              "0"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 6,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Multiplication by Constant\n",
        "diff(3*x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 7,
          "data": {
            "text/plain": [
              "3"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 7,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Power Rule\n",
        "diff(x**3)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 8,
          "data": {
            "text/plain": [
              "3*x**2"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 8,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Sum Rule\n",
        "diff(x**2+3*x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 9,
          "data": {
            "text/plain": [
              "2*x + 3"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 9,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Difference Rule\n",
        "diff(x**2-3*x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 10,
          "data": {
            "text/plain": [
              "2*x - 3"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 10,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Product Rule\n",
        "diff(x**2*x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 11,
          "data": {
            "text/plain": [
              "3*x**2"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 11,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Chain Rule\n",
        "diff(ln(x**2))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 12,
          "data": {
            "text/plain": [
              "2/x"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 12,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Example\n",
        "diff(9*(x+x**2))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 13,
          "data": {
            "text/plain": [
              "18*x + 9"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 13,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Matrix Calculus\n",
        "a = diff(3*x**2*y,x)\n",
        "b = diff(3*x**2*y,y)\n",
        "c = diff(2*x+8*y**7,x)\n",
        "d = diff(2*x+y**8,y)"
      ],
      "outputs": [],
      "execution_count": 14,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "print(a)\n",
        "print(b)\n",
        "print(c)\n",
        "print(d)"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "6*x*y\n",
            "3*x**2\n",
            "2\n",
            "8*y**7\n"
          ]
        }
      ],
      "execution_count": 15,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "Matrix_Calculus = np.matrix([[a, c], [b,d]])\n",
        "Matrix_Calculus "
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 16,
          "data": {
            "text/plain": [
              "matrix([[6*x*y, 2],\n",
              "        [3*x**2, 8*y**7]], dtype=object)"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 16,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Chain Rule\n",
        "diff(ln(sin(x**3)**2))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 17,
          "data": {
            "text/plain": [
              "6*x**2*cos(x**3)/sin(x**3)"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 17,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Sigmoid\n",
        "# d/dx S(x)=S(x)(1−S(x)) \n",
        "diff(1/(1+math.e**-x),x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 18,
          "data": {
            "text/plain": [
              "1.0*2.71828182845905**(-x)/(1 + 2.71828182845905**(-x))**2"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 18,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Integral\n",
        "import sympy\n",
        "sympy.init_printing()\n",
        "sympy.integrate(2*x, (x, 1, 0))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 19,
          "data": {
            "image/png": [
              "iVBORw0KGgoAAAANSUhEUgAAAB4AAAASCAYAAABM8m7ZAAAABHNCSVQICAgIfAhkiAAAAJVJREFUSInt1KENAjEUh/Ef5IZgAYbAcWEBBKMwADuwwAn2QLAADkVyCoUCgwNBFaLpXUpDyH3JSyte+4n+X/kTVtjigBue2JUQH4PsjlNJcY0pRpjHxFVm8T61cZxZnMwgbr0DkVq9E/sZrjMeHc5fcokXfS/qys+8cTFyfyDLUDAJ6wxN2F+xzuwEG/EpaL8hHYjyAqjXJ2KDRzE8AAAAAElFTkSuQmCC\n"
            ],
            "text/plain": [
              "-1"
            ],
            "text/latex": [
              "$$-1$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 19,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Jacobian\n",
        "from sympy import sin, cos, Matrix\n",
        "from sympy.abc import rho, phi\n",
        "\n",
        "X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])\n",
        "Y = Matrix([rho, phi])\n",
        "\nX.jacobian(Y)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 20,
          "data": {
            "text/plain": [
              "⎡cos(φ)  -ρ⋅sin(φ)⎤\n",
              "⎢                 ⎥\n",
              "⎢sin(φ)  ρ⋅cos(φ) ⎥\n",
              "⎢                 ⎥\n",
              "⎣ 2⋅ρ        0    ⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}\\cos{\\left (\\phi \\right )} & - \\rho \\sin{\\left (\\phi \\right )}\\\\\\sin{\\left (\\phi \\right )} & \\rho \\cos{\\left (\\phi \\right )}\\\\2 \\rho & 0\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 20,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Example Calculating the Jacobian\n",
        "# Equation\n",
        "eq = x**2*y + 3/4*x*y + 10\n",
        "eq"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 21,
          "data": {
            "image/png": [
              "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\n"
            ],
            "text/plain": [
              " 2                  \n",
              "x ⋅y + 0.75⋅x⋅y + 10"
            ],
            "text/latex": [
              "$$x^{2} y + 0.75 x y + 10$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 21,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Jacobian matrix \n",
        "x, y, z = symbols('x y z')\n",
        "Matrix([sin(x) + y, cos(y) + x, z]).jacobian([x, y, z])"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 22,
          "data": {
            "text/plain": [
              "⎡cos(x)     1     0⎤\n",
              "⎢                  ⎥\n",
              "⎢  1     -sin(y)  0⎥\n",
              "⎢                  ⎥\n",
              "⎣  0        0     1⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}\\cos{\\left (x \\right )} & 1 & 0\\\\1 & - \\sin{\\left (y \\right )} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 22,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# (x,y)=(0,0)\n",
        "x1 = -y\n",
        "x2 = x - 2*y * (2-x**2)\n",
        "J = sympy.Matrix([x1,x2])\n",
        "J.jacobian([x,y])"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 23,
          "data": {
            "text/plain": [
              "⎡    0         -1   ⎤\n",
              "⎢                   ⎥\n",
              "⎢              2    ⎥\n",
              "⎣4⋅x⋅y + 1  2⋅x  - 4⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}0 & -1\\\\4 x y + 1 & 2 x^{2} - 4\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 23,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "J.jacobian([x,y]).subs([(x,0), (y,0)])"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 24,
          "data": {
            "text/plain": [
              "⎡0  -1⎤\n",
              "⎢     ⎥\n",
              "⎣1  -4⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}0 & -1\\\\1 & -4\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 24,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Derivatives of x\n",
        "de_x = diff(x**2*y + 3/4*x*y + 10, x)\n",
        "de_x"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 25,
          "data": {
            "image/png": [
              "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\n"
            ],
            "text/plain": [
              "2⋅x⋅y + 0.75⋅y"
            ],
            "text/latex": [
              "$$2 x y + 0.75 y$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 25,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Derivatives of y\n",
        "de_y = diff(x**2*y + 3/4*x*y + 10, y)\n",
        "de_y"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 26,
          "data": {
            "image/png": [
              "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\n"
            ],
            "text/plain": [
              " 2         \n",
              "x  + 0.75⋅x"
            ],
            "text/latex": [
              "$$x^{2} + 0.75 x$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 26,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Example\n",
        "F = sympy.Matrix([de_x,de_y])\n",
        "F.jacobian([x,y])"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 27,
          "data": {
            "text/plain": [
              "⎡   2⋅y      2⋅x + 0.75⎤\n",
              "⎢                      ⎥\n",
              "⎣2⋅x + 0.75      0     ⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}2 y & 2 x + 0.75\\\\2 x + 0.75 & 0\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 27,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "F.jacobian([x,y]).subs([(x,0), (y,0)])"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 28,
          "data": {
            "text/plain": [
              "⎡ 0    0.75⎤\n",
              "⎢          ⎥\n",
              "⎣0.75   0  ⎦"
            ],
            "text/latex": [
              "$$\\left[\\begin{matrix}0 & 0.75\\\\0.75 & 0\\end{matrix}\\right]$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 28,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Hessian\n",
        "from sympy import Function, hessian, pprint\n",
        "from sympy.abc import x, y\n",
        "f = Function('f')(x, y)\n",
        "g1 = Function('g')(x, y)\n",
        "g2 = x**2+y**2\n",
        "pprint(hessian(f, (x,y), [g1, g2]))"
      ],
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "⎡                   ∂               ∂            ⎤\n",
            "⎢     0        0    ──(g(x, y))     ──(g(x, y))  ⎥\n",
            "⎢                   ∂x              ∂y           ⎥\n",
            "⎢                                                ⎥\n",
            "⎢     0        0        2⋅x             2⋅y      ⎥\n",
            "⎢                                                ⎥\n",
            "⎢                     2               2          ⎥\n",
            "⎢∂                   ∂               ∂           ⎥\n",
            "⎢──(g(x, y))  2⋅x   ───(f(x, y))   ─────(f(x, y))⎥\n",
            "⎢∂x                   2            ∂y ∂x         ⎥\n",
            "⎢                   ∂x                           ⎥\n",
            "⎢                                                ⎥\n",
            "⎢                     2               2          ⎥\n",
            "⎢∂                   ∂               ∂           ⎥\n",
            "⎢──(g(x, y))  2⋅y  ─────(f(x, y))   ───(f(x, y)) ⎥\n",
            "⎢∂y                ∂y ∂x              2          ⎥\n",
            "⎣                                   ∂y           ⎦\n"
          ]
        }
      ],
      "execution_count": 29,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def tanh(x):\n",
        "    return np.tanh(x)\n",
        "\n",
        "X = np.linspace(-5, 5, 100)\n",
        "plt.plot(X, tanh(X),'b')\n",
        "plt.xlabel('X Axis')\n",
        "plt.ylabel('Y Axis')\n",
        "plt.title('Neural Networks - Activation Function')\n",
        "plt.grid()\n",
        "plt.text(4, 0.8, r'$\\sigma(x)=\\tanh{(x)}}$', fontsize=16)\n",
        "plt.show()"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 30,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def sigma(x):\n",
        "    return (np.exp(x) - np.exp(-x)) / (np.exp(x) + np.exp(-x))\n",
        "\n",
        "X = np.linspace(-5, 5, 100)\n",
        "plt.plot(X, sigma(X),'b')\n",
        "plt.xlabel('X Axis')\n",
        "plt.ylabel('Y Axis')\n",
        "plt.title('Sigmoid Function')\n",
        "plt.grid()\n",
        "plt.text(4, 0.8, r'$\\sigma(x)=\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$', fontsize=16)\n",
        "plt.show()"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 31,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def derivatives_sigma(x):\n",
        "    return 1 / (np.cosh(x))**2\n",
        "\n",
        "X = np.linspace(-5, 5, 100)\n",
        "plt.plot(X, derivatives_sigma(X),'b')\n",
        "plt.xlabel('X Axis')\n",
        "plt.ylabel('Y Axis')\n",
        "plt.title('Sigmoid Function')\n",
        "plt.grid()\n",
        "plt.text(4, 0.8, r'$\\sigma(x)=\\frac{1}{cosh^2(x)}$', fontsize=16)\n",
        "plt.show()"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 32,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def derivatives_sigma(x):\n",
        "    return 4 / (np.exp(x)+np.exp(-x))**2\n",
        "\n",
        "X = np.linspace(-5, 5, 100)\n",
        "plt.plot(X, derivatives_sigma(X),'b')\n",
        "plt.xlabel('X Axis')\n",
        "plt.ylabel('Y Axis')\n",
        "plt.title('Sigmoid Function')\n",
        "plt.grid()\n",
        "plt.text(4, 0.8, r'$\\sigma(x)=\\frac{4}{(e^{x}+e^{-x})^2}$', fontsize=16)\n",
        "plt.show()"
      ],
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": [
              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\n"
            ],
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 33,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Newton_Raphson\n",
        "# f(x) - the function of the polynomial\n",
        "from sympy import *\n",
        "\n",
        "def f(x):\n",
        "    function = x**3 - x - 1\n",
        "    return function\n",
        "\n",
        "def derivative(x): #function to find the derivative of the polynomial\n",
        "    derivative = diff(f(x), x)\n",
        "    return derivative\n",
        "\n",
        "def Newton_Raphson(x):\n",
        "\treturn (x - (f(x) / derivative(x)))\n",
        "\nNewton_Raphson(x)"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 34,
          "data": {
            "image/png": [
              "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\n"
            ],
            "text/plain": [
              "     3        \n",
              "    x  - x - 1\n",
              "x - ──────────\n",
              "        2     \n",
              "     3⋅x  - 1 "
            ],
            "text/latex": [
              "$$x - \\frac{x^{3} - x - 1}{3 x^{2} - 1}$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 34,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Advanced Chain Rule\n",
        "# F(y)=ln(1−5y2+y3)\n",
        "from sympy import *\n",
        "\n",
        "y = symbols('y')\n",
        "F = symbols('F', cls=Function)\n",
        "\n\nDerivative(ln(1 - 5*y**2 + y**3)).doit()"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 35,
          "data": {
            "image/png": [
              "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAsCAYAAAD/709QAAAABHNCSVQICAgIfAhkiAAABBpJREFUaIHt2luoVFUcx/GPWioZRVp2o7LI6KIlYQRSNg9ZnSKCLkhB+JARWA8hPUWUD4UFgSEJQS+nelAq66EgorLLg4cgySQsyehqdhEFM8hL2cN/Tyz3mXNm9pw9zvawv7CYvdb82fu3/rPXWnvWb1MzLjgHH2ErvsDtfVUzjjgTV2THM/ETTuifnPHLFpzbbxG9YGKB2AdFIvZmZQi3jOHa83G8uGvLYiHewi84jDtHiFuG7/A3NuHaEjUU5jbcjNm4CE/hIC7v4lwzxDy7oDR1wQCeFHP3SIldLHTfj0uwGvtUbOTsxgNJ/Wcsz8XMw35cmtWn4BPc22NtIyX2U7yYa/sGK5N6J/3oCZNwNw5gbtL+GtbmYj/E89nxhOz7Fb0Ul9EqsZNxCHfl2tfg46Terh+lM1cMm0PYI6aGlOXYntTvEHf1jKx+Df7F5qSkP0yZtErsWVn7wlz749iW1Nv1o3Qm40Kx8KzELsxJvl8ghE8XQ/5bPNzFdVZk5xmtNNqcY7TE5herJ/B1Uh9zP44rEiyGfvOX/AxXZRdcmrVtymLm40qxSKwpeA1iyK1rE/NjF+fdhX9wRq59Jn5L6mPuR9HE5pmIqUl9Pz7HrViCezJRRdmVlbI5IJK2SMyjTRZhfVIvqx8d8bQYQrPEvLhSzJcDubhVWfu7vRIyCieK1XueGMqPZMfpo9RikeCl4nHrObFunJc711HrxyB+EL/m73gfN7aIWyIWt8t6LagFDa3n48Fc3DJ8L/qyyfDFjP72oyXv6W5erRqV6MdEnI5HsROn9FdO11SuHw0xH32l/L+pR5OG8dGPmpqamnHNhOzzcF9V1NgoTMAvxY5Q1ThmzcqTss9JwpqZ10ctraiMWVnE8yK8LmL7cHLJWspgp7hTib/de3BqP4QUTSxhbTT3CjaXK6flPuyvXZ6rF2YlHRqW3ST2apwtpoE5bWK7YZsY0s3SjcMwAy/jPp0vzIM6s4ymiVHx0GhBaWKLGGh7sQE3dSCkKIfEXdosfxTUOAVvim3NjT3Q9w4ewxujBaWJHRKOQMoq4Whuxck4LWufihscaWeUxQXYIXz/tWL/t1ONE8SdtwGv9EBbV7Qz0GaJvcsteve4NZBddy6uxwdiQZreocaxmJWDirvHo70U8j9VMgKbTBNeVHP4l6WR2Brcl5SDwllI29q9ITNiYlPPq4pG4F9iiM8uWSO8gFeT+jNiClqdtO3o8txHJLaKRuBUXCxelqBck293Vpr8mdW3tw4vRv5xa0i8/DaEt8u4QEGexXU4XzzWvS6mg5eSmH5rTA1LYu3JG5bD6LeBtk48eB8Qw3C94Y96vdI4qLPFq6Ezw/IIKmGgteFY0IgKGmgtOBY0DqOh+gZaQ/U11tTU1NTU1NS04D+oZ0OW+Cq4wAAAAABJRU5ErkJggg==\n"
            ],
            "text/plain": [
              "    2        \n",
              " 3⋅y  - 10⋅y \n",
              "─────────────\n",
              " 3      2    \n",
              "y  - 5⋅y  + 1"
            ],
            "text/latex": [
              "$$\\frac{3 y^{2} - 10 y}{y^{3} - 5 y^{2} + 1}$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 35,
      "metadata": {
        "collapsed": false,
        "outputHidden": false,
        "inputHidden": false
      }
    },
    {
      "cell_type": "code",
      "source": [
        "diff(ln(1 - 5*y**2 + y**3))"
      ],
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 36,
          "data": {
            "image/png": [
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            ],
            "text/plain": [
              "    2        \n",
              " 3⋅y  - 10⋅y \n",
              "─────────────\n",
              " 3      2    \n",
              "y  - 5⋅y  + 1"
            ],
            "text/latex": [
              "$$\\frac{3 y^{2} - 10 y}{y^{3} - 5 y^{2} + 1}$$"
            ]
          },
          "metadata": {}
        }
      ],
      "execution_count": 36,
      "metadata": {
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        "outputHidden": false,
        "inputHidden": false
      }
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      "display_name": "Python 3"
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    "language_info": {
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