01_Differentiation_Integration.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differentation & Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Some elementary functions and their derivatives:\n",
    "\n",
    "$$\n",
    "    \\cos (x)                           \\qquad \\frac{d}{dx} \\left( \\cos (x) \\right) = - \\sin(x)\n",
    "    \\\\\n",
    "    \\sin (x)                           \\qquad \\frac{d}{dx} \\left( \\sin (x) \\right) = \\cos (x)\n",
    "    \\\\\n",
    "    \\tan (x)                           \\qquad \\frac{d}{dx} \\left( \\tan (x)  \\right) = \\sec^2 (x)\n",
    "    \\\\\n",
    "    \\ln (x)                            \\qquad \\frac{d}{dx} \\left( \\ln (x)  \\right) = \\frac{1}{x}\n",
    "    \\\\\n",
    "    x^n + x^{n-1} + \\ldots + x + c     \n",
    "    \\qquad \\frac{d}{dx} \\left( x^n + x^{n-1} + \\ldots + x + c \\right) \n",
    "    =  (n)x^{n-1} + (n-1)x^{n-2} + \\ldots + 1 + 0\n",
    "$$\n",
    "\n",
    "Let's validate these with `sympy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sympy version is: \t1.3\n"
     ]
    }
   ],
   "source": [
    "# Setup\n",
    "from sympy import *\n",
    "from sympy import __version__ as sympy_version\n",
    "print(\"Sympy version is: \\t\" + sympy_version)\n",
    "x = Symbol('x')\n",
    "# from sympy.abc import x     # Prepare the variable x.\n",
    "init_printing()             # Make things print nicely"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = cos(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "(y_prime)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = sin(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\tan^{2}{\\left (x \\right )} + 1$$"
      ],
      "text/plain": [
       "   2       \n",
       "tan (x) + 1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = tan(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAqBAMAAAB1rqf/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiRInduzLEnmxuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXUlEQVQYGWNggAFGZRDLJOQzWICNGhSr03cPmPGk0//BgHR92HWYpczKTmBg3sBxjKeAgduA+zPjBAZmBvYLYNX8CWDq/QMgxWiQz8BmwLBe4D7DDGB4pJlFNDAAAN1MIGxCExJiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{x}$$"
      ],
      "text/plain": [
       "1\n",
       "─\n",
       "x"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = ln(x)\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$3 x^{2} + 2 x + 1$$"
      ],
      "text/plain": [
       "   2          \n",
       "3⋅x  + 2⋅x + 1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = x**3 + x**2 + x + 1\n",
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More complex derivatives\n",
    "\n",
    "Say we have a more difficult function to differentiate, using a symbolic package we can easily obtain an accurate and useful answer to our problem in a few lines of code.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (4 x \\right )} \\cos{\\left (3 x \\right )} \\tan{\\left (5 x \\right )}$$"
      ],
      "text/plain": [
       "sin(4⋅x)⋅cos(3⋅x)⋅tan(5⋅x)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = cos(3*x)*sin(4*x)*tan(5*x)\n",
    "\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left(5 \\tan^{2}{\\left (5 x \\right )} + 5\\right) \\sin{\\left (4 x \\right )} \\cos{\\left (3 x \\right )} - 3 \\sin{\\left (3 x \\right )} \\sin{\\left (4 x \\right )} \\tan{\\left (5 x \\right )} + 4 \\cos{\\left (3 x \\right )} \\cos{\\left (4 x \\right )} \\tan{\\left (5 x \\right )}$$"
      ],
      "text/plain": [
       "⎛     2         ⎞                                                             \n",
       "⎝5⋅tan (5⋅x) + 5⎠⋅sin(4⋅x)⋅cos(3⋅x) - 3⋅sin(3⋅x)⋅sin(4⋅x)⋅tan(5⋅x) + 4⋅cos(3⋅x\n",
       "\n",
       "                   \n",
       ")⋅cos(4⋅x)⋅tan(5⋅x)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_prime = diff(y)\n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Perhaps we don't know the coefficients?\n",
    "\n",
    "Still doable!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (b x \\right )} \\cos{\\left (a x \\right )} \\tan{\\left (c x \\right )}$$"
      ],
      "text/plain": [
       "sin(b⋅x)⋅cos(a⋅x)⋅tan(c⋅x)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import a,b,c\n",
    "y = cos(a*x)*sin(b*x)*tan(c*x)\n",
    "\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- a \\sin{\\left (a x \\right )} \\sin{\\left (b x \\right )} \\tan{\\left (c x \\right )} + b \\cos{\\left (a x \\right )} \\cos{\\left (b x \\right )} \\tan{\\left (c x \\right )} + c \\left(\\tan^{2}{\\left (c x \\right )} + 1\\right) \\sin{\\left (b x \\right )} \\cos{\\left (a x \\right )}$$"
      ],
      "text/plain": [
       "                                                                 ⎛   2        \n",
       "-a⋅sin(a⋅x)⋅sin(b⋅x)⋅tan(c⋅x) + b⋅cos(a⋅x)⋅cos(b⋅x)⋅tan(c⋅x) + c⋅⎝tan (c⋅x) + \n",
       "\n",
       " ⎞                  \n",
       "1⎠⋅sin(b⋅x)⋅cos(a⋅x)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Differentiate with respect to x. Optional second argument indicates what to differentiate with\n",
    "y_prime = diff(y, x)            \n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arbitrary Functions\n",
    "\n",
    "We can even manipulate functions that have not been explicitly specified with `sympy`. We do this with the `Function()` function.\n",
    "\n",
    "We can even do single and multi-variable functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import Symbol, Function\n",
    "\n",
    "x = Symbol('x')\n",
    "\n",
    "f = Function('f')(x)\n",
    "\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x,y \\right)}$"
      ],
      "text/plain": [
       "f(x, y)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import Symbol, Function\n",
    "\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "\n",
    "f = Function('f')(x,y)\n",
    "\n",
    "display(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then manipulate them with integration and differentiation. This is useful for validating transformation rules and how they affect function (whether a transformation is linear or not, how multiplication of functions affects differentiation, etc.)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int f{\\left(x \\right)}\\, dx$"
      ],
      "text/plain": [
       "Integral(f(x), x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d x} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "Derivative(f(x), x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import integrate, diff\n",
    "F = integrate(f,x)\n",
    "f_prime = diff(f,x)\n",
    "\n",
    "display(F)\n",
    "display(f_prime)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Repetitive Derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say we want to differentiate to a certain power, we can specify that with a third argument to `diff()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$a^{100} \\cos{\\left (a x \\right )}$$"
      ],
      "text/plain": [
       " 100         \n",
       "a   ⋅cos(a⋅x)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = cos(a*x)\n",
    "\n",
    "# Here, the third argument indicates how many times to differentiate.\n",
    "# We are taking the 100th derivative.\n",
    "y_prime = diff(y, x, 100)            \n",
    "\n",
    "y_prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a high-order derivative of a more complicated function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\frac{256 \\left(1382 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{5} \\tan{\\left (x \\right )} + 7192 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{4} \\tan^{3}{\\left (x \\right )} + 5097 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3} \\tan^{5}{\\left (x \\right )} + 502 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{7}{\\left (x \\right )} + 2 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{9}{\\left (x \\right )} - \\frac{310 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{5}}{x} - \\frac{5360 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{4} \\tan^{2}{\\left (x \\right )}}{x} - \\frac{7260 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3} \\tan^{4}{\\left (x \\right )}}{x} - \\frac{1235 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{6}{\\left (x \\right )}}{x} - \\frac{10 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{8}{\\left (x \\right )}}{x} + \\frac{2790 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{4} \\tan{\\left (x \\right )}}{x^{2}} + \\frac{8640 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3} \\tan^{3}{\\left (x \\right )}}{x^{2}} + \\frac{2700 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{5}{\\left (x \\right )}}{x^{2}} + \\frac{45 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{7}{\\left (x \\right )}}{x^{2}} - \\frac{765 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{4}}{x^{3}} - \\frac{8100 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3} \\tan^{2}{\\left (x \\right )}}{x^{3}} - \\frac{5130 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{4}{\\left (x \\right )}}{x^{3}} - \\frac{180 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{6}{\\left (x \\right )}}{x^{3}} + \\frac{5355 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3} \\tan{\\left (x \\right )}}{x^{4}} + \\frac{8190 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{3}{\\left (x \\right )}}{x^{4}} + \\frac{630 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{5}{\\left (x \\right )}}{x^{4}} - \\frac{1890 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{3}}{x^{5}} - \\frac{10395 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan^{2}{\\left (x \\right )}}{x^{5}} - \\frac{1890 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{4}{\\left (x \\right )}}{x^{5}} + \\frac{9450 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2} \\tan{\\left (x \\right )}}{x^{6}} + \\frac{4725 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{3}{\\left (x \\right )}}{x^{6}} - \\frac{4725 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)^{2}}{x^{7}} - \\frac{9450 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan^{2}{\\left (x \\right )}}{x^{7}} + \\frac{14175 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right) \\tan{\\left (x \\right )}}{x^{8}} - \\frac{14175 \\left(\\tan^{2}{\\left (x \\right )} + 1\\right)}{x^{9}} + \\frac{14175 \\tan{\\left (x \\right )}}{x^{10}}\\right)}{x}$$"
      ],
      "text/plain": [
       "    ⎛                                                                         \n",
       "    ⎜                  5                            4                         \n",
       "    ⎜     ⎛   2       ⎞                ⎛   2       ⎞     3           ⎛   2    \n",
       "256⋅⎜1382⋅⎝tan (x) + 1⎠ ⋅tan(x) + 7192⋅⎝tan (x) + 1⎠ ⋅tan (x) + 5097⋅⎝tan (x) \n",
       "    ⎜                                                                         \n",
       "    ⎝                                                                         \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                                              \n",
       "    3                            2                                         ⎛  \n",
       "   ⎞     5          ⎛   2       ⎞     7        ⎛   2       ⎞    9      310⋅⎝ta\n",
       "+ 1⎠ ⋅tan (x) + 502⋅⎝tan (x) + 1⎠ ⋅tan (x) + 2⋅⎝tan (x) + 1⎠⋅tan (x) - ───────\n",
       "                                                                              \n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "          5                     4                             3               \n",
       " 2       ⎞         ⎛   2       ⎞     2           ⎛   2       ⎞     4          \n",
       "n (x) + 1⎠    5360⋅⎝tan (x) + 1⎠ ⋅tan (x)   7260⋅⎝tan (x) + 1⎠ ⋅tan (x)   1235\n",
       "─────────── - ─────────────────────────── - ─────────────────────────── - ────\n",
       " x                         x                             x                    \n",
       "                                                                              \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "              2                                                        4      \n",
       " ⎛   2       ⎞     6         ⎛   2       ⎞    8           ⎛   2       ⎞       \n",
       "⋅⎝tan (x) + 1⎠ ⋅tan (x)   10⋅⎝tan (x) + 1⎠⋅tan (x)   2790⋅⎝tan (x) + 1⎠ ⋅tan(x\n",
       "─────────────────────── - ──────────────────────── + ─────────────────────────\n",
       "         x                           x                            2           \n",
       "                                                                 x            \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                      3                             2                         \n",
       "         ⎛   2       ⎞     3           ⎛   2       ⎞     5         ⎛   2      \n",
       ")   8640⋅⎝tan (x) + 1⎠ ⋅tan (x)   2700⋅⎝tan (x) + 1⎠ ⋅tan (x)   45⋅⎝tan (x) + \n",
       "─ + ─────────────────────────── + ─────────────────────────── + ──────────────\n",
       "                  2                             2                           2 \n",
       "                 x                             x                           x  \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                              4                     3                         \n",
       " ⎞    7          ⎛   2       ⎞         ⎛   2       ⎞     2           ⎛   2    \n",
       "1⎠⋅tan (x)   765⋅⎝tan (x) + 1⎠    8100⋅⎝tan (x) + 1⎠ ⋅tan (x)   5130⋅⎝tan (x) \n",
       "────────── - ────────────────── - ─────────────────────────── - ──────────────\n",
       "                      3                         3                             \n",
       "                     x                         x                             x\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                    x                                         \n",
       "\n",
       "    2                                                         3               \n",
       "   ⎞     4          ⎛   2       ⎞    6           ⎛   2       ⎞                \n",
       "+ 1⎠ ⋅tan (x)   180⋅⎝tan (x) + 1⎠⋅tan (x)   5355⋅⎝tan (x) + 1⎠ ⋅tan(x)   8190⋅\n",
       "───────────── - ───────────────────────── + ────────────────────────── + ─────\n",
       "3                            3                           4                    \n",
       "                            x                           x                     \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "             2                                                         3      \n",
       "⎛   2       ⎞     3          ⎛   2       ⎞    5           ⎛   2       ⎞       \n",
       "⎝tan (x) + 1⎠ ⋅tan (x)   630⋅⎝tan (x) + 1⎠⋅tan (x)   1890⋅⎝tan (x) + 1⎠    103\n",
       "────────────────────── + ───────────────────────── - ─────────────────── - ───\n",
       "         4                            4                        5              \n",
       "        x                            x                        x               \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                2                                                          2  \n",
       "   ⎛   2       ⎞     2           ⎛   2       ⎞    4           ⎛   2       ⎞   \n",
       "95⋅⎝tan (x) + 1⎠ ⋅tan (x)   1890⋅⎝tan (x) + 1⎠⋅tan (x)   9450⋅⎝tan (x) + 1⎠ ⋅t\n",
       "───────────────────────── - ────────────────────────── + ─────────────────────\n",
       "           5                             5                            6       \n",
       "          x                             x                            x        \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                       2                      \n",
       "             ⎛   2       ⎞    3           ⎛   2       ⎞         ⎛   2       ⎞ \n",
       "an(x)   4725⋅⎝tan (x) + 1⎠⋅tan (x)   4725⋅⎝tan (x) + 1⎠    9450⋅⎝tan (x) + 1⎠⋅\n",
       "───── + ────────────────────────── - ─────────────────── - ───────────────────\n",
       "                     6                         7                        7     \n",
       "                    x                         x                        x      \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "                                                                         ⎞\n",
       "   2            ⎛   2       ⎞                ⎛   2       ⎞               ⎟\n",
       "tan (x)   14175⋅⎝tan (x) + 1⎠⋅tan(x)   14175⋅⎝tan (x) + 1⎠   14175⋅tan(x)⎟\n",
       "─────── + ────────────────────────── - ─────────────────── + ────────────⎟\n",
       "                       8                         9                10     ⎟\n",
       "                      x                         x                x       ⎠\n",
       "──────────────────────────────────────────────────────────────────────────\n",
       "                                                                          "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = tan(x)*1/x\n",
    "y_prime = diff(y, x, 10)\n",
    "\n",
    "y_prime   # Note that large functions format poorly on github, but nicely when run on locally on your computer (hopefully)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, computer-based symbolic manipulations show their strength."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration\n",
    "\n",
    "[Link](https://docs.sympy.org/1.3/modules/integrals/integrals.html?highlight=integrate#sympy.integrals.integrate) \n",
    "to the documentation for `integrate` for `sympy` version 1.3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sympy version is: \t1.3\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "from sympy.abc import x\n",
    "from sympy import __version__ as sympy_version\n",
    "print(\"Sympy version is: \\t\" + str(sympy_version))\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simple example\n",
    "Indefinite integral."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = cos(x)\n",
    "Y = integrate(y)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definite integral.\n",
    "\n",
    "See how we pass a `tuple` as the second argument to indicate to the `integrate()` function what to integrate with respect to and with what bounds. \n",
    "\n",
    "It's formatted like: `integrate(function, (variable, lower_limit, upper_limit))`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIpm7MhCriUTv3c12VLgeopIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABOSURBVAgdY2BUMnZgYAhjYH/BwJDKwDCTgWEWA0OeA8O+ABAJBOsCgATHcxCTKwFEKoEIHgUQeYmBUYCBRYGBR4BBqrwoi4Fh37t3rxgAK5QOlzv7snYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = sin(x)\n",
    "attributes = (x, 0, pi)\n",
    "Y = integrate(y, attributes)     # Integrate sin(x) from 0 to pi\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A more complicated example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{e^{x} \\sin{\\left (x \\right )}}{2} + \\frac{e^{x} \\cos{\\left (x \\right )}}{2}$$"
      ],
      "text/plain": [
       " x           x       \n",
       "ℯ ⋅sin(x)   ℯ ⋅cos(x)\n",
       "───────── + ─────────\n",
       "    2           2    "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = cos(x)*exp(x)\n",
    "Y = integrate(y)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perhaps with some coefficients?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$a \\left(\\frac{3 e^{4 x} \\sin{\\left (3 x \\right )}}{25} + \\frac{4 e^{4 x} \\cos{\\left (3 x \\right )}}{25}\\right)$$"
      ],
      "text/plain": [
       "  ⎛   4⋅x               4⋅x         ⎞\n",
       "  ⎜3⋅ℯ   ⋅sin(3⋅x)   4⋅ℯ   ⋅cos(3⋅x)⎟\n",
       "a⋅⎜─────────────── + ───────────────⎟\n",
       "  ⎝       25                25      ⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import a\n",
    "y = a*cos(3*x)*exp(4*x)\n",
    "Y = integrate(y, x)\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$a \\log{\\left (a \\right )} - a + 1$$"
      ],
      "text/plain": [
       "a⋅log(a) - a + 1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y = integrate(log(x), (x, 1, a))\n",
    "\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# -------- Differentation Questions --------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1:\n",
    "\n",
    "Calculate the second derivative of:\n",
    "\n",
    "$$\n",
    "\\frac\n",
    "{\\tan (x) + \\sin(x)}\n",
    "{x}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 1:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2: (Requires the knowledge of `plotting`)\n",
    "\n",
    "Consider the function:\n",
    "\n",
    "$$\n",
    "    \\sqrt{x} \\cdot \\sin\\left(x\\right) \n",
    "$$\n",
    "\n",
    "Symbolically plot it in blue(`'b'`)  and its derivative in red (`'r'`) on the same plot over the domain $[0,2\\pi]$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3:\n",
    "\n",
    "\n",
    "If $n$ is the 3rd number in your Student ID number, find the $n$-th derivative of $y = e^x \\cdot \\sin(x)$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 4:\n",
    "\n",
    "If $f$ is a function of $x$ and $g$ is a function of $x$, find:\n",
    "\n",
    "$$\n",
    "    \\frac{d}{dx} \\left( f\\cdot g \\right) \n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$f{\\left (x \\right )} \\frac{d}{d x} g{\\left (x \\right )} + g{\\left (x \\right )} \\frac{d}{d x} f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "     d               d       \n",
       "f(x)⋅──(g(x)) + g(x)⋅──(f(x))\n",
       "     dx              dx      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 4:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 5:\n",
    "\n",
    "### (i)\n",
    "If $f$ is a function of $x$ and $g$ is a function of $x$, find:\n",
    "\n",
    "$$\n",
    "    \\frac{d}{dx} \\left( \\frac{f}{g} \\right) \n",
    "$$\n",
    "\n",
    "### (ii)\n",
    "How does this stack up with your knowledge of the quotient rule? Is the answer equivalent?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 6:\n",
    "\n",
    "Differentiate the following function four times.\n",
    "\n",
    "$$ f(x) = \\frac{1}{(x^2+x+1)^2} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 6:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# -------- Integration Questions --------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 1:\n",
    "\n",
    "Evaluate:\n",
    "\n",
    "$$\n",
    "\\int^{\\pi / 2}_0 \\sin^2(x) \\cos^3(x) dx\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 1:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 2:\n",
    "\n",
    "In the cell below (after you evaluate it with `Shift+Enter`) you will have a function $f(r) = 2\\pi r$\n",
    "\n",
    "### (i)\n",
    "What is the formula $2\\pi r$ used to calculate?\n",
    "\n",
    "### (ii)\n",
    "Integrate this function, print and interpret the result. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Question 3:\n",
    "\n",
    "Evaluate the following repeated integral (all integrations are done with respect to $x$).\n",
    "\n",
    "$$\n",
    "\\iiint \\frac{dx^3 }{(x+2)^2 (x+3)^3}\n",
    "$$\n",
    "\n",
    "A note on repeated integrals [here](http://mathworld.wolfram.com/RepeatedIntegral.html), which discusses better (and more rigorous ways) of representing them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Question 4:\n",
    "\n",
    "Make a python function that takes in a mathematical function to be integrated $n$ times with respect to some variable, $x$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Answer 3:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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