diff --git a/math/note05.ipynb b/math/note05.ipynb
index a4021d58..fa2a19f0 100644
--- a/math/note05.ipynb
+++ b/math/note05.ipynb
@@ -49,11 +49,11 @@
     "\n",
     "#### Example: First Derivative as a Matrix\n",
     "\n",
-    "Suppose we discretize a function $\\psi(x)$ over a set of points $\\{x_1, x_2, \\dots, x_n\\}$ with spacing $h$. We can approximate the first derivative using the finite difference formula:\n",
+    "- Suppose we discretize a function $\\psi(x)$ over a set of points $\\{x_1, x_2, \\dots, x_n\\}$ with spacing $h$. We can approximate the first derivative using the **finite difference**:\n",
     "\n",
     "$$ \\frac{d\\psi}{dx} \\approx \\frac{\\psi(x_{i+1}) - \\psi(x_i)}{h} $$\n",
     "\n",
-    "This operation can be represented by a matrix. For $n = 5$ grid points, the first derivative matrix $D_1$ looks like:\n",
+    "- This operation can be represented by a matrix. For $n = 5$ grid points, the first derivative matrix  $D_1$ in finite difference apporxmation looks like:\n",
     "\n",
     "$$\n",
     "D_1 = \\frac{1}{h}\n",
@@ -66,7 +66,7 @@
     "\\end{pmatrix}\n",
     "$$\n",
     "\n",
-    "For a function represented as a vector $\\psi = (\\psi(x_1), \\psi(x_2), \\dots, \\psi(x_5))$, applying this matrix gives the derivative at each point (except the boundary)."
+    "- For a function represented as a vector $\\psi = (\\psi(x_1), \\psi(x_2), \\dots, \\psi(x_5))$, applying this matrix gives the derivative at each point (except the boundary)."
    ]
   },
   {
@@ -113,7 +113,7 @@
     "\\end{pmatrix}\n",
     "$$\n",
     "\n",
-    "This approximates the first derivative of \\( \\psi(x) = x^2 \\) at each of the sampled points:\n",
+    "This approximates the first derivative of $\\psi(x) = x^2$ at each of the sampled points:\n",
     "\n",
     "- At $x = 1$, the derivative is approximately 1 (exact value: 2).\n",
     "- At $x = 2$, the derivative is approximately 3 (exact value: 4).\n",
@@ -150,7 +150,57 @@
     "# Apply the first derivative matrix to the function vector\n",
     "derivative_psi = D_1 @ psi\n",
     "\n",
-    "print('derivative of x^2', derivative_psi)  "
+    "print('derivative of x^2 using finite difference matrix', derivative_psi)  \n",
+    "\n",
+    "# Exact derivative of psi = x^2 is 2*x\n",
+    "exact_derivative_psi = 2 * x\n",
+    "print('\\nExact derivative of x^2:\\n', exact_derivative_psi)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Centered Difference approximation\n",
+    "\n",
+    "- A much better approximation is obtained by using centered difference. \n",
+    "- And taking care of boundary points via higher order evaluation of derivatives. \n",
+    "\n",
+    "$$ \\frac{d\\psi}{dx} \\approx \\frac{\\psi(x_{i+1}) - \\psi(x_{i-1})}{2h} $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "# Define the grid points and function\n",
+    "n = 5\n",
+    "x = np.linspace(0, 4, n)  # Grid points from 0 to 4\n",
+    "h = x[1] - x[0]  # Spacing between points\n",
+    "psi = x**2  # Function psi(x) = x^2\n",
+    "\n",
+    "# Define the centered difference approximation for the first derivative matrix\n",
+    "D_1_centered = (1/(2*h)) * np.array([\n",
+    "    [-3, 4, -1, 0, 0],  # One-sided difference for the first point (improved boundary handling)\n",
+    "    [-1, 0, 1, 0, 0],   # Centered difference for interior points\n",
+    "    [0, -1, 0, 1, 0],\n",
+    "    [0, 0, -1, 0, 1],\n",
+    "    [0, 0, 1, -4, 3]    # One-sided difference for the last point (improved boundary handling)\n",
+    "])\n",
+    "\n",
+    "# Apply the centered difference matrix to the function vector\n",
+    "derivative_psi_centered = D_1_centered @ psi\n",
+    "\n",
+    "# Print the results\n",
+    "print('First derivative using centered difference approximation:\\n', derivative_psi_centered)\n",
+    "\n",
+    "# Exact derivative of psi = x^2 is 2*x\n",
+    "exact_derivative_psi = 2 * x\n",
+    "print('\\nExact derivative of x^2:\\n', exact_derivative_psi)"
    ]
   },
   {
@@ -242,7 +292,7 @@
     "# Calculate the second derivative of x^2 using the centered difference method\n",
     "second_derivative_centered = D_2_centered @ psi\n",
     "\n",
-    "print('second derivative of x^2 by using centered difference \\n', D_1 @ D_1@x)"
+    "print('second derivative of x^2 by using centered difference \\n', second_derivative_centered)"
    ]
   },
   {