Magnetorquers Partial Derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "N_d\n",
    "=\n",
    "\\frac{\n",
    "    4\\left[\n",
    "        \\ln\\left(\n",
    "            \\frac{l_c}{r_c}\n",
    "        \\right)\n",
    "        -1\n",
    "    \\right]\n",
    "}{\n",
    "    \\left(\n",
    "        \\frac{l_c}{r_c}\n",
    "    \\right)\n",
    "    ^2-4\n",
    "    \\ln\\left(\n",
    "        \\frac{l_c}{r_c}\n",
    "    \\right)\n",
    "}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "M\n",
    "=\n",
    "\\frac{r_c V}{2 W_w}\n",
    "\\left[\n",
    "    1+\n",
    "    \\frac{\n",
    "        \\mu_r-1\n",
    "    }{\n",
    "        1+N_d(\\mu_r-1)\n",
    "    }\n",
    "\\right]\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For simplicity in this notebook, we will use all of the above variables without subscripts, as displayed below:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "N\n",
    "=\n",
    "\\frac{\n",
    "    4\\left[\n",
    "        \\ln\\left(\n",
    "            l/r\n",
    "        \\right)\n",
    "        -1\n",
    "    \\right]\n",
    "}{\n",
    "    \\left(\n",
    "        l/r\n",
    "    \\right)\n",
    "    ^2-4\n",
    "    \\ln\\left(\n",
    "        l/r\n",
    "    \\right)\n",
    "}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    M\n",
    "    & =\n",
    "    \\frac{r V}{2 W}\n",
    "    \\left[\n",
    "        1+\n",
    "        \\frac{\n",
    "            \\mu-1\n",
    "        }{\n",
    "            1+N(\\mu-1)\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To begin finding the partial derivatives of $N$, we will start by breaking up $N$ into two functions. This will make calculating the partial derivatives via the quotient rule more straight forward."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "N(l,r)\n",
    "=\n",
    "\\frac{f(r,l)}{g(r,l)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    f(l,r)\n",
    "    & =\n",
    "    4\n",
    "    \\left[\n",
    "      \\ln\\left(\n",
    "          \\frac{l}{r}\n",
    "      \\right)\n",
    "      -1\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    4\n",
    "    \\ln\\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    -4\n",
    "    \\\\\n",
    "    & = \n",
    "    4 \\ln l -4 \\ln r - 4\n",
    "\\\\\\\\\n",
    "    g(l,r)\n",
    "    & =\n",
    "    \\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    ^2-4\n",
    "    \\ln\\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{l^2}{r^2}\n",
    "    - 4 \\ln l + 4 \\ln r \n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now also calculate the partial derivatives of $f(l,r)$ and $g(l,r)$, as well as $(g(l,r))^2$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    f_l(l,r)\n",
    "    & = \n",
    "    \\frac{4}{l}\n",
    "    \\\\\n",
    "    f_r(l,r)\n",
    "    & =\n",
    "    -\n",
    "    \\frac{4}{r}\n",
    "    \\\\\n",
    "    g_l(l,r)\n",
    "    & =\n",
    "    \\frac{2l}{r^2}\n",
    "    -\n",
    "    \\frac{4}{l}\n",
    "    \\\\\n",
    "    g_r(l,r)\n",
    "    & =\n",
    "    -\n",
    "    \\frac{2l^2}{r^3}\n",
    "    +\n",
    "    \\frac{4}{r}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    (g(l,r))^2\n",
    "    & =\n",
    "    \\left(\n",
    "        \\frac{l^2}{r^2}\n",
    "        - 4\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "    \\right)\n",
    "    ^2\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{l^4}{r^4}\n",
    "    -\n",
    "    \\frac{8 l^2}{r^2}\n",
    "    \\ln\\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    + 16\n",
    "    \\left(\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "    \\right)\n",
    "    ^2\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{l^4}{r^4}\n",
    "    -\n",
    "    \\frac{8 l^2}{r^2}\n",
    "    \\ln\\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    +\n",
    "    16\n",
    "    \\left(\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "    \\right)\n",
    "    ^2\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{l^4}{r^4}\n",
    "    -\n",
    "    \\frac{8 l^2 r^2}{r^4}\n",
    "    \\ln\\left(\n",
    "        \\frac{l}{r}\n",
    "    \\right)\n",
    "    +\n",
    "    \\frac{16 r^4}{r^4}\n",
    "    \\left(\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "    \\right)\n",
    "    ^2\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        l^4\n",
    "        -\n",
    "        8 l^2 r^2\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "        +\n",
    "        16 r^4\n",
    "        \\left(\n",
    "            \\ln\\left(\n",
    "                \\frac{l}{r}\n",
    "            \\right)\n",
    "        \\right)\n",
    "        ^2\n",
    "    }{\n",
    "        r^4\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4r^2\n",
    "            \\ln\\left(\n",
    "                \\frac{l}{r}\n",
    "            \\right)\n",
    "        \\right)\n",
    "        ^2\n",
    "    }{\n",
    "        r^4\n",
    "    }\n",
    "    =\n",
    "    \\left(\n",
    "        l^2\n",
    "        -\n",
    "        4r^2\n",
    "        \\ln\\left(\n",
    "            \\frac{l}{r}\n",
    "        \\right)\n",
    "    \\right)\n",
    "    ^2\n",
    "    r^{-4}\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From here, we can now calculate the partial derivatives of $N$ using the quotient rule. To assist with clarity, $q=\\ln\\left(\\frac{l}{r}\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    N_l\n",
    "    &=\n",
    "    \\frac{\n",
    "        f_l(l,r) g(l,r) - f(l,r) g_l(l,r)\n",
    "    }{\n",
    "        (g(l,r))^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            \\frac{4}{l}\n",
    "        \\right)\n",
    "        \\left(\n",
    "            \\frac{l^2}{r^2}\n",
    "            - 4 q\n",
    "        \\right)\n",
    "        -\n",
    "        \\left(\n",
    "            4 q\n",
    "            - 4\n",
    "        \\right)\\left(\n",
    "            \\frac{2 l}{r^2}\n",
    "            -\n",
    "            \\frac{4}{l}\n",
    "        \\right)\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\frac{4 l}{r^2}\n",
    "        -\n",
    "        \\frac{16 q}{l}\n",
    "        -\n",
    "        \\frac{8 l q}{r^2}\n",
    "        +\n",
    "        \\frac{16 q}{l}\n",
    "        +\n",
    "        \\frac{8 l}{r^2}\n",
    "        -\n",
    "        \\frac{16}{l}\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\frac{\n",
    "            12l^2\n",
    "            -8l^2q\n",
    "            +16r^2\n",
    "        }{\n",
    "            l r^2\n",
    "        }\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        4r^2\n",
    "        \\left(\n",
    "            3l^2\n",
    "            +4r^2\n",
    "            -2l^2q\n",
    "        \\right)\n",
    "    }{\n",
    "        l\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        4r^2\n",
    "        \\left[\n",
    "            3 l^2\n",
    "            +4 r^2\n",
    "            -2 l^2\n",
    "            \\ln\\left(\n",
    "                \\frac{l}{r}\n",
    "            \\right)\n",
    "        \\right]\n",
    "    }{\n",
    "        l\n",
    "        \\left[\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2\n",
    "            \\ln\\left(\n",
    "                \\frac{l}{r}\n",
    "            \\right)\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    N_r\n",
    "    &=\n",
    "    \\frac{\n",
    "        f_r(l,r) g(l,r) - f(l,r) g_r(l,r)\n",
    "    }{\n",
    "        (g(l,r))^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            -\n",
    "            \\frac{4}{r}\n",
    "        \\right)\n",
    "        \\left(\n",
    "            \\frac{l^2}{r^2}\n",
    "            - 4 q\n",
    "        \\right)\n",
    "        -\n",
    "        \\left(\n",
    "            4 q\n",
    "            - 4\n",
    "        \\right)\n",
    "        \\left(\n",
    "            -\n",
    "            \\frac{2 l^2}{r^3}\n",
    "            +\n",
    "            \\frac{4}{r}\n",
    "        \\right)\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        - \\frac{4 l^2}{r^3}\n",
    "        + \\frac{16 q}{r}\n",
    "        + \\frac{8 l^2 q}{r^3}\n",
    "        - \\frac{16 q}{r}\n",
    "        - \\frac{8 l^2}{r^3}\n",
    "        + \\frac{16}{r}\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        - \\frac{12 l^2}{r^3}\n",
    "        + \\frac{8 l^2 q}{r^3}\n",
    "        + \\frac{16 r^2}{r^3}\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "        r^{-4}\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        4 r\n",
    "        \\left(\n",
    "           -3l^2 + 4r^2 + 2l^2q\n",
    "        \\right)\n",
    "    }{\n",
    "        \\left(\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2 q\n",
    "        \\right)\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        4 r\n",
    "        \\left[\n",
    "           -3l^2 + 4r^2 + 2l^2\n",
    "           \\ln\\left(\n",
    "               \\frac{l}{r}\n",
    "           \\right)\n",
    "        \\right]\n",
    "    }{\n",
    "        \\left[\n",
    "            l^2\n",
    "            -\n",
    "            4 r^2\n",
    "            \\ln\\left(\n",
    "               \\frac{l}{r}\n",
    "           \\right)\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With the partial derivatives of $N$, we can now find the partial derivatives for $M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    M\n",
    "    & =\n",
    "    \\frac{r V}{2 W}\n",
    "    \\left[\n",
    "        1+\n",
    "        \\frac{\n",
    "            \\mu-1\n",
    "        }{\n",
    "            1+N(\\mu-1)\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{r V}{2W}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            1 + (\\mu - 1) N\n",
    "        }{\n",
    "            1 + (\\mu - 1) N\n",
    "        }\n",
    "        +\n",
    "        \\frac{\n",
    "            \\mu - 1\n",
    "        }{\n",
    "            1 + (\\mu - 1) N\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{r V}{2W}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + (\\mu - 1) N\n",
    "        }{\n",
    "            1 + (\\mu - 1) N\n",
    "        }\n",
    "    \\right]\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will split up the quotient found in the brackets here to calculate its partial derivatives. This is a necessary step to help calculate the partial derivatives of $M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    f(l,r)\n",
    "    & =\n",
    "    \\mu + \\left(\\mu - 1\\right) N\n",
    "    \\\\\n",
    "    f_l(l,r)\n",
    "    & =\n",
    "    \\left(\\mu - 1\\right) N_l\n",
    "    \\\\\n",
    "    f_r(l,r)\n",
    "    & =\n",
    "    \\left(\\mu - 1\\right) N_r\n",
    "    \\\\\n",
    "    g(l,r)\n",
    "    & =\n",
    "    1 + \\left(\\mu - 1\\right) N\n",
    "    \\\\\n",
    "    g_l(l,r)\n",
    "    & =\n",
    "    \\left(\\mu - 1\\right) N_l\n",
    "    \\\\\n",
    "    g_r(l,r)\n",
    "    & =\n",
    "    \\left(\\mu - 1\\right) N_r\n",
    "    \\\\\n",
    "    (g(l,r))^2\n",
    "    & =\n",
    "    \\left(1 + \\left(\\mu - 1\\right) N\\right)^2\n",
    "    \\\\\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    \\frac{\\partial}{\\partial l}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    & =\n",
    "    \\frac{\n",
    "        f_l(l,r) g(l,r) - f(l,r) g_l(l,r)\n",
    "    }{\n",
    "        (g(l,r))^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left[\n",
    "            \\left(\\mu - 1 \\right) N_l\n",
    "        \\right]\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1 \\right) N\n",
    "        \\right]\n",
    "        -\n",
    "        \\left[\n",
    "            \\mu + \\left(\\mu - 1 \\right) N\n",
    "        \\right]\n",
    "        \\left[\n",
    "            \\left(\\mu - 1 \\right) N_l\n",
    "        \\right]\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            \\mu - 1\n",
    "        \\right)\n",
    "        N_l\n",
    "        +\n",
    "        \\left(\n",
    "            \\mu - 1\n",
    "        \\right)\n",
    "        ^2 N N_l\n",
    "        -\n",
    "        \\left(\n",
    "            \\mu^2 - \\mu\n",
    "        \\right)\n",
    "        N_l\n",
    "        -\n",
    "        \\left(\n",
    "            \\mu - 1\n",
    "        \\right)\n",
    "        ^2 N N_l\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            \\mu - 1 - \\mu^2 + \\mu\n",
    "        \\right)\n",
    "        N_l\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "    =\n",
    "    \\frac{\n",
    "        \\left(\n",
    "            - \\mu^2 + 2 \\mu - 1\n",
    "        \\right)\n",
    "        N_l\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right]\n",
    "        ^2\n",
    "    }\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        - \\left(\\mu - 1\\right)^2 N_l\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right] ^2\n",
    "    }\n",
    "    \\\\\n",
    "    \\frac{\\partial}{\\partial r}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    & =\n",
    "    \\frac{\n",
    "        - \\left(\\mu - 1\\right)^2 N_r\n",
    "    }{\n",
    "        \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right] ^2\n",
    "    }\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From here, we can now calculate the partial derivatives of $M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    M_l\n",
    "    & =\n",
    "    \\frac{\\partial}{\\partial l}\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    +\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\frac{\\partial}{\\partial l}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    0\n",
    "    +\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            - \\left(\\mu - 1\\right)^2 N_l\n",
    "        }{\n",
    "            \\left[\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right] ^2\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{\n",
    "        - r V \\left(\\mu - 1\\right)^2 N_l\n",
    "    }{\n",
    "        2 W \\left[\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        \\right] ^2\n",
    "    }\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{align}\n",
    "    M_r\n",
    "    & =\n",
    "    \\frac{\\partial}{\\partial r}\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    +\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\frac{\\partial}{\\partial r}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\left(\n",
    "        \\frac{V}{2 W}\n",
    "    \\right)\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "    \\right]\n",
    "    +\n",
    "    \\left(\n",
    "        \\frac{r V}{2 W}\n",
    "    \\right)\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            - \\left(\\mu - 1\\right)^2 N_r\n",
    "        }{\n",
    "            \\left[\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right] ^2\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{V}{2 W}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\mu + \\left(\\mu - 1\\right) N\n",
    "        }{\n",
    "            1 + \\left(\\mu - 1\\right) N\n",
    "        }\n",
    "        +\n",
    "        \\frac{\n",
    "            -r \\left(\\mu - 1\\right)^2 N_r\n",
    "        }{\n",
    "            \\left[\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right] ^2\n",
    "        }\n",
    "    \\right]\n",
    "    \\\\\n",
    "    & =\n",
    "    \\frac{V}{2 W}\n",
    "    \\left[\n",
    "        \\frac{\n",
    "            \\left(\n",
    "                \\mu + \\left(\\mu - 1\\right) N\n",
    "            \\right)\n",
    "            \\left(\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right)\n",
    "        }{\n",
    "            \\left[\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right] ^2\n",
    "        }\n",
    "        +\n",
    "        \\frac{\n",
    "            -r \\left(\\mu - 1\\right)^2 N_r\n",
    "        }{\n",
    "            \\left[\n",
    "                1 + \\left(\\mu - 1\\right) N\n",
    "            \\right] ^2\n",
    "        }\n",
    "    \\right]\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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