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  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "# Automatic Differentiation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-07-03T21:50:54.398998Z",
     "start_time": "2019-07-03T21:50:53.307986Z"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 2., 3.])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from mxnet import autograd, np, npx  \n",
    "npx.set_np()\n",
    "\n",
    "x = np.arange(4)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Allocate space to store the gradient with respect to ``x``."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-07-03T21:50:54.404851Z",
     "start_time": "2019-07-03T21:50:54.401388Z"
    },
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   },
   "outputs": [],
   "source": [
    "x.attach_grad()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Record the computation within the `record` scope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-07-03T21:50:54.416246Z",
     "start_time": "2019-07-03T21:50:54.406690Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "4"
    },
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array(28.)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "with autograd.record():\n",
    "    y = 2.0 * np.dot(x, x)\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "The gradient of the function $y = 2\\mathbf{x}^{\\top}\\mathbf{x}$ with respect to $\\mathbf{x}$ should be $4\\mathbf{x}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2019-07-03T21:50:54.435955Z",
     "start_time": "2019-07-03T21:50:54.418269Z"
    },
    "attributes": {
     "classes": [],
     "id": "",
     "n": "5"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 0., 0., 0.])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.backward()\n",
    "x.grad - 4 * x"
   ]
  }
 ],
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