notebooks/sage-3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Differentials\n",
    "\n",
    "Sage helps in taking differentials of expressions and functions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(a, b, c, x)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var('a,b,c,x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivatives with single variable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ a x^{5} + b x^{3} + c x</script></html>"
      ],
      "text/plain": [
       "x |--> a*x^5 + b*x^3 + c*x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f(x) = a*x^5 + b*x^3 + c*x\n",
    "f.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ 5 \\, a x^{4} + 3 \\, b x^{2} + c</script></html>"
      ],
      "text/plain": [
       "x |--> 5*a*x^4 + 3*b*x^2 + c"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "df(x)=f.derivative(x)\n",
    "df.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x \\cos\\left(x^{2}\\right) e^{\\left(\\sin\\left(x^{2}\\right)\\right)}</script></html>"
      ],
      "text/plain": [
       "2*x*cos(x^2)*e^(sin(x^2))"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "diff(e^sin(x^2),x).show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{x^{2} + 1}</script></html>"
      ],
      "text/plain": [
       "1/(x^2 + 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "diff(arctan(x),x).show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Visualizing functions and derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 1 graphics primitive"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(x) = x^3 - x + 2\n",
    "plot(g(x),-2,2,color='blue')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can take a derivative of the function and superpose the plots to understand the variations in the slope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gprime(x) = g(x).derivative()\n",
    "fig1 = plot(g(x),-2,2,color='blue')\n",
    "fig2 = plot(gprime(x),-2,2,color='red')\n",
    "fig = fig1 + fig2\n",
    "fig.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Partial derivatives of multivariable functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 3 \\, x^{4} y^{3} + 9 \\, x^{2} y - 4 \\, x + 8 \\, y</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 3*x^4*y^3 + 9*x^2*y - 4*x + 8*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p(x,y) = 3*x^4*y^3 + 9*y*x^2 - 4*x + 8*y\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we evaluate $ {\\partial p \\over \\partial x} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 12 \\, x^{3} y^{3} + 18 \\, x y - 4</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 12*x^3*y^3 + 18*x*y - 4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dpdx(x,y) = diff(p,x)\n",
    "dpdx.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we evaluate $ {\\partial p \\over \\partial y} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 9 \\, x^{4} y^{2} + 9 \\, x^{2} + 8</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 9*x^4*y^2 + 9*x^2 + 8"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dpdy(x,y)=diff(p,y)\n",
    "dpdy.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One can check if the order of differentiation matters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we evaluate $ {\\partial^2 p \\over \\partial x \\partial y} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 36 \\, x^{3} y^{2} + 18 \\, x</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 36*x^3*y^2 + 18*x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d2pdxy(x,y)=diff(p,x,y)\n",
    "d2pdxy.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we evaluate $ {\\partial^2 p \\over \\partial y \\partial x} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 36 \\, x^{3} y^{2} + 18 \\, x</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 36*x^3*y^2 + 18*x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d2pdyx(x,y)=diff(p,y,x)\n",
    "d2pdyx.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Double differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we evaluate $ {\\partial^2 p \\over \\partial x^2} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( x, y \\right) \\ {\\mapsto} \\ 36 \\, x^{2} y^{3} + 18 \\, y</script></html>"
      ],
      "text/plain": [
       "(x, y) |--> 36*x^2*y^3 + 18*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d2pdx2(x,y)=diff(p,x,2)\n",
    "d2pdx2.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implicit differentiation\n",
    "\n",
    "In situations when a function is not defined explicity but we need to evaluate its derivative, implicit differentiation is useful. For example:\n",
    "\n",
    "Given\n",
    "$$ 5 h(x)^2 + \\sin(h) = x^2 $$\n",
    "Evaluate $ {\\partial h \\over \\partial x}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}x \\ {\\mapsto}\\ 5 \\, h\\left(x\\right)^{2} + \\sin\\left(h\\left(x\\right)\\right) = x^{2}</script></html>"
      ],
      "text/plain": [
       "x |--> 5*h(x)^2 + sin(h(x)) == x^2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "h(x) = function('h')(x)\n",
    "expr = 5*h^2 + sin(h) == x^2\n",
    "expr.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Uncomment the lines in the two cells below and run those in your notebook locally. Commenting the cells here to help github render the notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "#dhdx = solve(diff(expr),diff(h))\n",
    "#dhdx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "#dhdx[0].show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.0",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}