Example notebook demonstrating differentiation

with an actual problem taken from math.stackoverflow.com

Author: abinashmeher999

notebooks/differentiation_example.ipynb

@@ -0,0 +1,179 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Differentiation example\n",
+    "This demonstrates solving the problem I found [here](http://math.stackexchange.com/questions/221197/a-tough-differential-calculus-problem), using the *SymEngine* gem.\n",
+    "\n",
+    "---"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The function is described as \n",
+    "$$f_p(x) = \\dfrac{9\\sqrt{x^2+p}}{x^2+2}$$\n",
+    "We need to differentiate w.r.t. $x$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "One solution is:\n",
+    "$$f_p^2(x) (x^2 + 2)^2 = 81(x^2 + p) \\quad\\Longrightarrow$$\n",
+    "\n",
+    "$$2 f_p(x) f_p'(x)(x^2 + 2)^2 + 4x f_p^2(x^2 +2) = 162x \\quad\\Longrightarrow$$\n",
+    "\n",
+    "\\begin{align}f_p'(x) &= \\frac{162x - 4x f_p^2(x)(x^2+2)}{2f_p(x)(x^2+2)^2}\\\\ \\\\ &= \\frac{162x - 324x\\frac{x^2 + p}{x^2 +2}}{18(x^2 +2)\\sqrt{x^2+p}}\\\\ \\\\ &= \\frac{9 x (x^2+2) - 18x(x^2+p)}{(x^2 + 2)^2 \\sqrt{x^2 + p}}\\\\ \\\\ &= \\frac{9x(2-2p-x^2)}{(x^2 + 2)^2\\sqrt{x^2 + p}}\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We will attempt to solve this using features in *SymEngine*\n",
+    "\n",
+    "---"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "true"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "require 'symengine'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Declare the symbols"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(1/2)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "x = SymEngine::Symbol.new('x')\n",
+    "p = SymEngine::Symbol.new('p')\n",
+    "half = Rational('1/2')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Create the expression"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\"9*(p + x**2)**(1/2)/(2 + x**2)\""
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "fp = 9*((x**2 + p)**half)/((x**2)+2)\n",
+    "fp.to_s"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Differentiate wrt $x$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\"-18*x*(p + x**2)**(1/2)/(2 + x**2)**2 + 9*x/((2 + x**2)*(p + x**2)**(1/2))\""
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "answer = fp.diff(x)\n",
+    "answer.to_s"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Which is indeed correct! If you simplify the second last answer that we got from our solution. You will find the exact same answer. Using *SymEngine* for solving problems is as simple as that."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Ruby 2.2.0",
+   "language": "ruby",
+   "name": "ruby"
+  },
+  "language_info": {
+   "file_extension": "rb",
+   "mimetype": "application/x-ruby",
+   "name": "ruby",
+   "version": "2.2.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}