{
 "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."
   ]
  }
 ],
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