diff --git a/Rapport.ipynb b/Rapport.ipynb
index 712180305eea0bbf0a47b607cdea243a3c44f393..39ebf1fa035be593b43aaf0bb58e3f4ac5e46031 100644
--- a/Rapport.ipynb
+++ b/Rapport.ipynb
@@ -473,102 +473,122 @@
     "\n",
     "### 1\n",
     "\n",
-    "$$\n",
-    "\\begin{align*}\n",
-    "\\sigma(x)&=\\frac{1}{1+e^{-x}} \\\\\n",
-    "\\Rightarrow \\sigma'(x)&=\\frac{-e^{-x}}{-(1+e^{-x})^2} \\\\\n",
-    "\\Rightarrow \\sigma'(x)&=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}}) \\\\\n",
-    "\\Rightarrow \\sigma'(x)&=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}}) \\\\\n",
-    "\\Rightarrow \\sigma'(x)&=\\sigma(1-\\sigma)\n",
-    "\\end{align*}\n",
-    "$$\n",
+    "$$\\sigma(x)=\\frac{1}{1+e^{-x}} $$\n",
+    "$$\\Rightarrow \\sigma'(x)=\\frac{-e^{-x}}{-(1+e^{-x})^2} $$\n",
+    "$$\\Rightarrow \\sigma'(x)=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}}) $$\n",
+    "$$\\Rightarrow \\sigma'(x)=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}}) $$\n",
+    "$$\\Rightarrow \\sigma'(x)=\\sigma(1-\\sigma)$$\n",
     "\n",
     "### 2\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial A^{(2)}} ? \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}&=\\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (a_i^{(2)} - y_i)^2 \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}&=\\frac{2}{N_{out}} (a_i^{(2)} - y_i) \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(2)}}&=\\frac{2}{N_{out}} (A^{(2)} - Y)\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial A^{(2)}} ? $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}=\\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (a_i^{(2)} - y_i)^2 $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial a_i^{(2)}}=\\frac{2}{N_{out}} (a_i^{(2)} - y_i) $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(2)}}=\\frac{2}{N_{out}} (A^{(2)} - Y)\n",
+    "\n",
     "$$\n",
     "\n",
     "### 3\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial Z^{(2)}}&=\\frac{\\partial C}{\\partial A^{(2)}}\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}} \\\\\n",
-    "A^{(2)}&=\\sigma (Z^{(2)}) \\\\\n",
-    "\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}}&=A^{(2)}(1-A^{(2)}) \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(2)}}&=\\frac{\\partial C}{\\partial A^{(2)}}[A^{(2)}(1-A^{(2)})]\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial Z^{(2)}}=\\frac{\\partial C}{\\partial A^{(2)}}\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}} $$\n",
+    "$$\n",
+    "A^{(2)}=\\sigma (Z^{(2)}) $$\n",
+    "$$\n",
+    "\\frac{\\partial A^{(2)}}{\\partial Z^{(2)}}=A^{(2)}(1-A^{(2)}) $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(2)}}=\\frac{\\partial C}{\\partial A^{(2)}}[A^{(2)}(1-A^{(2)})]\n",
+    "\n",
     "$$\n",
     "\n",
     "### 4\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial W^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}} \\\\\n",
-    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}}&=A^{(1)} \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}A^{(1)}\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial W^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}} $$\n",
+    "$$\n",
+    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)} $$\n",
+    "$$\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial W^{(2)}}=A^{(1)} $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}A^{(1)}\n",
+    "\n",
     "$$\n",
     "\n",
     "### 5\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial B^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}} \\\\\n",
-    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}}&=1 \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(2)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial B^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}} $$\n",
+    "$$\n",
+    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)} $$\n",
+    "$$\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial B^{(2)}}=1 $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(2)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\n",
+    "\n",
     "$$\n",
     "\n",
     "### 6\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial A^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}} \\\\\n",
-    "Z^{(2)}&=W^{(2)}A^{(1)}+B^{(2)} \\\\\n",
-    "\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}}&=W^{(2)} \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(2)}}W^{(2)}\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial A^{(1)}}=\\frac{\\partial C}{\\partial Z^{(2)}}\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}} $$\n",
+    "$$\n",
+    "Z^{(2)}=W^{(2)}A^{(1)}+B^{(2)} $$\n",
+    "$$\n",
+    "\\frac{\\partial Z^{(2)}}{\\partial A^{(1)}}=W^{(2)} $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial A^{(1)}}=\\frac{\\partial C}{\\partial Z^{(2)}}W^{(2)}\n",
+    "\n",
     "$$\n",
     "\n",
     "### 7\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial Z^{(1)}}&=\\frac{\\partial C}{\\partial A^{(1)}}\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}} \\\\\n",
-    "A^{(1)}&=\\sigma (Z^{(1)}) \\\\\n",
-    "\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}}&=A^{(1)}(1-A^{(1)}) \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(1)}}&=\\frac{\\partial C}{\\partial A^{(1)}}[A^{(1)}(1-A^{(1)})]\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial Z^{(1)}}=\\frac{\\partial C}{\\partial A^{(1)}}\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}} $$\n",
+    "$$\n",
+    "A^{(1)}=\\sigma (Z^{(1)}) $$\n",
+    "$$\n",
+    "\\frac{\\partial A^{(1)}}{\\partial Z^{(1)}}=A^{(1)}(1-A^{(1)}) $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial Z^{(1)}}=\\frac{\\partial C}{\\partial A^{(1)}}[A^{(1)}(1-A^{(1)})]\n",
+    "\n",
     "$$\n",
     "\n",
     "### 8\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial W^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}} \\\\\n",
-    "Z^{(1)}&=W^{(1)}A^{(0)}+B^{(1)} \\\\\n",
-    "\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}}&=A^{(0)} \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}A^{(0)}\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial W^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}} $$\n",
+    "$$\n",
+    "Z^{(1)}=W^{(1)}A^{(0)}+B^{(1)} $$\n",
+    "$$\n",
+    "\\frac{\\partial Z^{(1)}}{\\partial W^{(1)}}=A^{(0)} $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial W^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}A^{(0)}\n",
+    "\n",
     "$$\n",
     "\n",
     "### 9\n",
     "\n",
     "$$\n",
-    "\\begin{align*}\n",
-    "\\frac{\\partial C}{\\partial B^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}} \\\\\n",
-    "Z^{(1)}&=W^{(1)}A^{(0)}+B^{(1)} \\\\\n",
-    "\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}}&=1 \\\\\n",
-    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(1)}}&=\\frac{\\partial C}{\\partial Z^{(1)}}\n",
-    "\\end{align*}\n",
+    "\n",
+    "\\frac{\\partial C}{\\partial B^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}} $$\n",
+    "$$\n",
+    "Z^{(1)}=W^{(1)}A^{(0)}+B^{(1)} $$\n",
+    "$$\n",
+    "\\frac{\\partial Z^{(1)}}{\\partial B^{(1)}}=1 $$\n",
+    "$$\n",
+    "\\Rightarrow \\frac{\\partial C}{\\partial B^{(1)}}=\\frac{\\partial C}{\\partial Z^{(1)}}\n",
+    "\n",
     "$$\n"
    ]
   },