{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4f4968c5-06cd-4c35-8acc-c0e25c395ab3",
   "metadata": {},
   "source": [
    "# Using derivative module\n",
    "This contains a small script on how to perform multi-point first and\n",
    "second numercial derivates comparing them to the expected result. Here,\n",
    "we try to approximate the first and second derivatives of $y = ln(x)$\n",
    "with a displacement of 0.1, 0.5, 1.0, and 1.5. We then show the\n",
    "improvement of the numercial derivative as we start to include more\n",
    "data points to the derivative. We have implemented code to calculate\n",
    "the numerical derivative with two, four, six, and eight symmetric\n",
    "displacment points. For the second derivative there is an extra point\n",
    "as the data from the equilibrium position is required. We use $y=ln(x)$\n",
    "as it has a very simple analytical first and second derivative of $1/x$\n",
    "and $-1/x^2$, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b7f09c30-843c-42ec-8c13-159e3073a993",
   "metadata": {},
   "outputs": [],
   "source": [
    "from vibrav.numerical.derivatives import (two_point_1d, four_point_1d, six_point_1d, eight_point_1d,\n",
    "                                          two_point_2d, four_point_2d, six_point_2d, eight_point_2d)\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ef37246a-a449-415d-8e21-3b24722ed820",
   "metadata": {},
   "outputs": [],
   "source": [
    "def do_derivs(delta, base, steps, stdout=False, first=True):\n",
    "    x = np.linspace(base-delta*steps/2,base+delta*steps/2,steps+1)\n",
    "    y = np.log(x)\n",
    "    y_plus = y[-int(steps/2):]\n",
    "    y_minus = np.flip(y[:int(steps/2)])\n",
    "    y_equil = y[int(steps/2)]\n",
    "    if first:\n",
    "        actual = 1/base\n",
    "        d_eight = eight_point_1d(y_plus[:4], y_minus[:4], delta)\n",
    "        d_six = six_point_1d(y_plus[:3], y_minus[:3], delta)\n",
    "        d_four = four_point_1d(y_plus[:2], y_minus[:2], delta)\n",
    "        d_two = two_point_1d(y_plus[:1], y_minus[:1], delta)\n",
    "    else:\n",
    "        actual = -1/base**2\n",
    "        d_eight = eight_point_2d(y_plus[:4], y_minus[:4], y_equil, delta)\n",
    "        d_six = six_point_2d(y_plus[:3], y_minus[:3], y_equil, delta)\n",
    "        d_four = four_point_2d(y_plus[:2], y_minus[:2], y_equil, delta)\n",
    "        d_two = two_point_2d(y_plus[:1], y_minus[:1], y_equil, delta)\n",
    "    cols = ['actual', 'delta']+[x+'-point' for x in ['two', 'four', 'six', 'eight']]\n",
    "    df = pd.Series([actual, delta, d_two, d_four, d_six, d_eight], index=cols)\n",
    "    return df\n",
    "\n",
    "def gen_plot(fig_num, deltas, base, steps):\n",
    "    fig = plt.figure(fig_num, figsize=(8,8), dpi=100)\n",
    "    for i, delta in enumerate(deltas):\n",
    "        ax = fig.add_subplot(2, 2, i+1)\n",
    "        x = np.linspace(base-delta*steps/2,base+delta*steps/2,steps+1)\n",
    "        y = np.log(x)\n",
    "        ax.axvline(base, color='k', linewidth=0.7)\n",
    "        ax.plot(x, y, label=\"Delta: {:.1f}\".format(delta))\n",
    "        act_x = np.linspace(base-delta*steps/2,base+delta*steps/2,1000)\n",
    "        act_y = np.log(act_x)\n",
    "        ax.plot(act_x, act_y, label=\"Actual\")\n",
    "        ax.legend(frameon=False, loc='upper left')\n",
    "        ax.text(0.52, 0.05, 'x = {:.2f}'.format(base), va='bottom', ha='left',\n",
    "                transform=ax.transAxes, fontweight='bold')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f8703a1c-41b6-4293-9768-28d64ee4df39",
   "metadata": {},
   "source": [
    "## Setting $x=7.8$\n",
    "\\begin{equation}\n",
    "\\frac{d}{dx}~ln(7.8) = 0.128205 ~~~~~~~~ \\frac{d^2}{dx^2}~ln(7.8) = -0.016437\n",
    "\\end{equation}\n",
    "### Plots with the numerical vs. actual values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "471eaa34-acf1-42b6-9083-8a37478b78f7",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 800x800 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gen_plot(1, [0.1, 0.5, 1, 1.5], 7.8, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1c2556f-8f63-4a8a-bf7a-49b052fc6cb5",
   "metadata": {},
   "source": [
    "### Numerical first derivative for delta of 0.1, 0.5, 1.0, and 1.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d867530b-1eb5-446f-b6f0-916d87a5ff37",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.128205</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.128212</td>\n",
       "      <td>0.128205</td>\n",
       "      <td>0.128205</td>\n",
       "      <td>0.128205</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.128205</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.128381</td>\n",
       "      <td>0.128203</td>\n",
       "      <td>0.128205</td>\n",
       "      <td>0.128205</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.128205</td>\n",
       "      <td>1.0</td>\n",
       "      <td>0.128915</td>\n",
       "      <td>0.128176</td>\n",
       "      <td>0.128209</td>\n",
       "      <td>0.128204</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.128205</td>\n",
       "      <td>1.5</td>\n",
       "      <td>0.129822</td>\n",
       "      <td>0.128044</td>\n",
       "      <td>0.128258</td>\n",
       "      <td>0.128155</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0  0.128205    0.1   0.128212    0.128205   0.128205     0.128205      0\n",
       "1  0.128205    0.5   0.128381    0.128203   0.128205     0.128205      1\n",
       "2  0.128205    1.0   0.128915    0.128176   0.128209     0.128204      2\n",
       "3  0.128205    1.5   0.129822    0.128044   0.128258     0.128155      3"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_1d = []\n",
    "for d, delta in enumerate([0.1, 0.5, 1, 1.5]):\n",
    "    df = do_derivs(delta, base=7.8, steps=10)\n",
    "    df['index'] = d\n",
    "    derivs_1d.append(df)\n",
    "derivs_1d = pd.concat(derivs_1d, axis=1).T\n",
    "derivs_1d['index'] = derivs_1d['index'].astype(int)\n",
    "derivs_1d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2c7f0f4-e734-454f-a593-24e2243de355",
   "metadata": {},
   "source": [
    "### Numerical second derivative for delta of 0.1, 0.5, 1.0, and 1.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "cb71d099-5281-44d7-9d81-5a76ff860512",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.016437</td>\n",
       "      <td>0.1</td>\n",
       "      <td>-0.016438</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.016437</td>\n",
       "      <td>0.5</td>\n",
       "      <td>-0.016470</td>\n",
       "      <td>-0.016436</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.016437</td>\n",
       "      <td>1.0</td>\n",
       "      <td>-0.016573</td>\n",
       "      <td>-0.016430</td>\n",
       "      <td>-0.016437</td>\n",
       "      <td>-0.016436</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.016437</td>\n",
       "      <td>1.5</td>\n",
       "      <td>-0.016748</td>\n",
       "      <td>-0.016402</td>\n",
       "      <td>-0.016449</td>\n",
       "      <td>-0.016425</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0 -0.016437    0.1  -0.016438   -0.016437  -0.016437    -0.016437      0\n",
       "1 -0.016437    0.5  -0.016470   -0.016436  -0.016437    -0.016437      1\n",
       "2 -0.016437    1.0  -0.016573   -0.016430  -0.016437    -0.016436      2\n",
       "3 -0.016437    1.5  -0.016748   -0.016402  -0.016449    -0.016425      3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_2d = []\n",
    "for d, delta in enumerate([0.1, 0.5, 1.0, 1.5]):\n",
    "    df = do_derivs(delta, base=7.8, steps=10, first=False)\n",
    "    df['index'] = d\n",
    "    derivs_2d.append(df)\n",
    "derivs_2d = pd.concat(derivs_2d, axis=1).T\n",
    "derivs_2d['index'] = derivs_2d['index'].astype(int)\n",
    "derivs_2d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f894f560-ead5-4893-b864-50848c6e3373",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Setting $x=2.51$\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{d}{dx}~ln(2.51) = 0.398406 ~~~~~~~~ \\frac{d^2}{dx^2}~ln(2.51) = -0.158728\n",
    "\\end{equation}\n",
    "### Plots with the numerical vs. actual values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "36fb22e9-2b8c-4b8c-b785-6504fc2bc170",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x800 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gen_plot(2, [0.1, 0.5], 2.8, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9625a38-0cba-450d-92b7-4653f67f9765",
   "metadata": {},
   "source": [
    "### Numerical first derivative for delta of 0.1, and 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "dd470084-3721-43ae-9296-7d9ab150d355",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.398406</td>\n",
       "      <td>0.1</td>\n",
       "      <td>0.398617</td>\n",
       "      <td>0.398406</td>\n",
       "      <td>0.398406</td>\n",
       "      <td>0.398406</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.398406</td>\n",
       "      <td>0.5</td>\n",
       "      <td>0.403805</td>\n",
       "      <td>0.397823</td>\n",
       "      <td>0.398617</td>\n",
       "      <td>0.398172</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0  0.398406    0.1   0.398617    0.398406   0.398406     0.398406      0\n",
       "1  0.398406    0.5   0.403805    0.397823   0.398617     0.398172      1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_1d = []\n",
    "for d, delta in enumerate([0.1, 0.5]):\n",
    "    df = do_derivs(delta, base=2.51, steps=10)\n",
    "    df['index'] = d\n",
    "    derivs_1d.append(df)\n",
    "derivs_1d = pd.concat(derivs_1d, axis=1).T\n",
    "derivs_1d['index'] = derivs_1d['index'].astype(int)\n",
    "derivs_1d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cfa3e52-f645-48ed-b570-bd4924553f2b",
   "metadata": {},
   "source": [
    "### Numerical second derivative for delta of 0.1, and 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "89e812ee-bec4-487d-b896-31bc60660abc",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-0.158728</td>\n",
       "      <td>0.1</td>\n",
       "      <td>-0.158854</td>\n",
       "      <td>-0.158727</td>\n",
       "      <td>-0.158728</td>\n",
       "      <td>-0.158728</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>-0.158728</td>\n",
       "      <td>0.5</td>\n",
       "      <td>-0.161963</td>\n",
       "      <td>-0.158337</td>\n",
       "      <td>-0.158877</td>\n",
       "      <td>-0.158555</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0 -0.158728    0.1  -0.158854   -0.158727  -0.158728    -0.158728      0\n",
       "1 -0.158728    0.5  -0.161963   -0.158337  -0.158877    -0.158555      1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_2d = []\n",
    "for d, delta in enumerate([0.1, 0.5]):\n",
    "    df = do_derivs(delta, base=2.51, steps=10, first=False)\n",
    "    df['index'] = d\n",
    "    derivs_2d.append(df)\n",
    "derivs_2d = pd.concat(derivs_2d, axis=1).T\n",
    "derivs_2d['index'] = derivs_2d['index'].astype(int)\n",
    "derivs_2d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a678b8d9-b350-44e3-8aef-d556df05cf9b",
   "metadata": {},
   "source": [
    "## Setting $x=0.51$\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{d}{dx}~ln(0.51) = 1.960784 ~~~~~~~~ \\frac{d^2}{dx^2}~ln(0.51) = -3.844675\n",
    "\\end{equation}\n",
    "### Plots with the numerical vs. actual values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "410c062a-debc-4f57-8991-4e6aff260bc4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gen_plot(3, [0.1], 0.51, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3754db1a-5dd2-4f13-be33-0ce93e9bf38d",
   "metadata": {},
   "source": [
    "### Numerical first derivative for delta of 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "666a296c-ffb4-4db2-a466-d4e1d8c80adb",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1.960784</td>\n",
       "      <td>0.1</td>\n",
       "      <td>1.986509</td>\n",
       "      <td>1.958101</td>\n",
       "      <td>1.961712</td>\n",
       "      <td>1.959824</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0  1.960784    0.1   1.986509    1.958101   1.961712     1.959824      0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_1d = []\n",
    "for d, delta in enumerate([0.1]):\n",
    "    df = do_derivs(delta, base=0.51, steps=10)\n",
    "    df['index'] = d\n",
    "    derivs_1d.append(df)\n",
    "derivs_1d = pd.concat(derivs_1d, axis=1).T\n",
    "derivs_1d['index'] = derivs_1d['index'].astype(int)\n",
    "derivs_1d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d434553e-ced3-4735-a083-83116e2764a5",
   "metadata": {},
   "source": [
    "### Numerical second derivative for delta of 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "ce3c9473-a33f-42a7-bd14-5d4fe508204b",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>actual</th>\n",
       "      <th>delta</th>\n",
       "      <th>two-point</th>\n",
       "      <th>four-point</th>\n",
       "      <th>six-point</th>\n",
       "      <th>eight-point</th>\n",
       "      <th>index</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>-3.844675</td>\n",
       "      <td>0.1</td>\n",
       "      <td>-3.920533</td>\n",
       "      <td>-3.835843</td>\n",
       "      <td>-3.847904</td>\n",
       "      <td>-3.841204</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     actual  delta  two-point  four-point  six-point  eight-point  index\n",
       "0 -3.844675    0.1  -3.920533   -3.835843  -3.847904    -3.841204      0"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "derivs_2d = []\n",
    "for d, delta in enumerate([0.1]):\n",
    "    df = do_derivs(delta, base=0.51, steps=10, first=False)\n",
    "    df['index'] = d\n",
    "    derivs_2d.append(df)\n",
    "derivs_2d = pd.concat(derivs_2d, axis=1).T\n",
    "derivs_2d['index'] = derivs_2d['index'].astype(int)\n",
    "derivs_2d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8719ad92-2e13-425a-987b-8864d6e50cde",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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