Chapter-11.ipynb

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   "source": [
    "%matplotlib widget\n",
    "from sympy import *\n",
    "init_printing(use_latex=True)"
   ]
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   "cell_type": "markdown",
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   "source": [
    "# 11 - Derivatives"
   ]
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   "source": [
    "## Explicit Differentiation"
   ]
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   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "f = x**3 + 2 * x**2 + x + 1\n",
    "diff(f, x)"
   ]
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   "source": [
    "diff(f, x, x, x)"
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   "source": [
    "diff(f, x, 3)"
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   "outputs": [],
   "source": [
    "y = symbols(\"y\")\n",
    "f = 3 * x**5 * y**2 + 5 * x + 2 * y + 6 * y**3 * x\n",
    "diff(f, x, y, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "diff(f, x, y, 2)"
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   "cell_type": "code",
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   "source": [
    "uneval_der = Derivative(f, x, y, 2)\n",
    "uneval_der"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "uneval_der.doit()"
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   "cell_type": "code",
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    "f.diff(x, 2, y, evaluate=False)"
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Function(\"f\")(x, y)\n",
    "der = f.diff(x, y, 2)\n",
    "der"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr = 3 * x**5 * y**2 + 5 * x + 2 * y + 6 * y**3 * x\n",
    "der = der.subs(f, expr)\n",
    "der"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "der.doit()"
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   "cell_type": "code",
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   "outputs": [],
   "source": [
    "der.args"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implicit Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x, y, z\")\n",
    "idiff(x**2 + y**2 - 1, y, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = idiff(x**2 + y + z - 1, [y, z], x, 2)\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = Function(\"y\")(x)\n",
    "z = Function(\"z\")(x)\n",
    "(x**2 + y + z - 1).diff(x, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Advanced Topics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "expr = x**2 + x * sin(x)\n",
    "display(expr, expr.diff(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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