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    "# Numerical Differentiation\n",
    "\n",
    "Finding a derivative of tabular data can be done using a finite\n",
    "difference. Here we essentially pick two points on a function or a set\n",
    "of data points and calculate the slope from there. Let’s imagine a\n",
    "domain $x$ as a vector such that $\\vec{x}$ =\n",
    "$\\pmatrix{x_0, x_1, x_2, ...}$. Then we can use the following methods to\n",
    "approximate derivatives\n",
    "\n",
    "## Forward Difference\n",
    "\n",
    "Uses the point at which we want to find the derivative and a point\n",
    "forwards on the line. $$\n",
    "f'(x_i) = \\frac{f(x_{i+1})-f(x_i)}{x_{i+1}-x_i}\n",
    "$$ *Hint: Consider what happens at the last point.*"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "d2fe3ad1-8af0-499a-8954-41a88a49b834",
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    {
     "ename": "IndexError",
     "evalue": "index 99 is out of bounds for axis 0 with size 99",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mIndexError\u001b[0m                                Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[2], line 10\u001b[0m\n\u001b[1;32m      6\u001b[0m y \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m34\u001b[39m \u001b[38;5;241m*\u001b[39m np\u001b[38;5;241m.\u001b[39mexp(\u001b[38;5;241m3\u001b[39m \u001b[38;5;241m*\u001b[39m x)\n\u001b[1;32m      8\u001b[0m dydx \u001b[38;5;241m=\u001b[39m  (y[\u001b[38;5;241m1\u001b[39m:] \u001b[38;5;241m-\u001b[39m y[:\u001b[38;5;241m-\u001b[39m\u001b[38;5;241m1\u001b[39m]) \u001b[38;5;241m/\u001b[39m (x[\u001b[38;5;241m1\u001b[39m:] \u001b[38;5;241m-\u001b[39m x[:\u001b[38;5;241m-\u001b[39m\u001b[38;5;241m1\u001b[39m])\n\u001b[0;32m---> 10\u001b[0m dydx[\u001b[38;5;28mlen\u001b[39m(y)]\u001b[38;5;241m=\u001b[39mdydx[\u001b[38;5;28mlen\u001b[39m(dydx)]\n\u001b[1;32m     12\u001b[0m \u001b[38;5;66;03m# Plot the function\u001b[39;00m\n\u001b[1;32m     13\u001b[0m plt\u001b[38;5;241m.\u001b[39mplot(x, y, label\u001b[38;5;241m=\u001b[39m\u001b[38;5;124mr\u001b[39m\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m$y(x)$\u001b[39m\u001b[38;5;124m'\u001b[39m)\n",
      "\u001b[0;31mIndexError\u001b[0m: index 99 is out of bounds for axis 0 with size 99"
     ]
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   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Initiate vectors\n",
    "x = np.linspace(0, 2, 100) \n",
    "y = 34 * np.exp(3 * x)\n",
    "\n",
    "dydx =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])\n",
    "\n",
    "dydx[len(y)]=dydx[len(dydx)]\n",
    "\n",
    "# Plot the function\n",
    "plt.plot(x, y, label=r'$y(x)$')\n",
    "plt.plot(x, dydx, label=r'$\\frac{dy}{dx}$')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Plot of $34e^{3x}$')\n",
    "plt.grid(True)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
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   "cell_type": "markdown",
   "id": "a54ad04f-05e5-4589-a633-d6dea2a5acfc",
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   "source": [
    "## Backwards Difference\n",
    "\n",
    "Uses the point at which we want to find $$\n",
    "f'(x_i) = \\frac{f(x_{i})-f(x_{i-1})}{x_i - x_{i-1}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "415d0b3c-a70d-48ef-93ad-6e827127b222",
   "metadata": {},
   "outputs": [],
   "source": [
    "dydx =  (y[1:] - y[:-1]) / (x[1:] - x[:-1])\n",
    "\n",
    "# Plot the function\n",
    "plt.plot(x, y, label=r'$y(x)$')\n",
    "plt.plot(x, dydx, label=b'$/frac{dy}{dx}$')\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.title('Plot of $34e^{3x}$')\n",
    "plt.grid(True)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21c87595-add2-4f7a-8d10-2da3b4284e02",
   "metadata": {},
   "source": [
    "## Central Difference\n",
    "\n",
    "$$\n",
    "f'(x_i) = \\frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6a67e63c-9516-415d-87d6-97d9d986ba33",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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