Update pages/Vanishing Gradient (Sigmoid).py

pages/Vanishing Gradient (Sigmoid).py

@@ -74,4 +74,42 @@ st.markdown("""
 **Highlighted Issue:**  
 - Repeated multiplication of small derivatives ($\\sigma'(z)$)  
 - Gradients shrink exponentially → **Vanishing Gradient Problem**
-""")
+""")
+
+import numpy as np
+import pandas as pd
+
+st.write("""
+The **vanishing gradient problem** occurs when gradients become very small during backpropagation,
+causing weights to update very slowly (or stop updating).
+This mostly happens with activation functions like **Sigmoid** or **tanh** in deep networks.
+""")
+
+# --- Define sigmoid & its derivative ---
+def sigmoid(x):
+    return 1 / (1 + np.exp(-x))
+
+def sigmoid_derivative(x):
+    s = sigmoid(x)
+    return s * (1 - s)
+
+# --- Generate values ---
+x = np.linspace(-10, 10, 200)
+y_sigmoid = sigmoid(x)
+y_grad = sigmoid_derivative(x)
+
+# --- Put into DataFrame for st.line_chart ---
+df = pd.DataFrame({
+    "x": x,
+    "Sigmoid": y_sigmoid,
+    "Gradient": y_grad
+})
+
+st.subheader("📊 Sigmoid Function vs Gradient")
+st.line_chart(df.set_index("x"))
+
+st.write("""
+- Notice how the **Sigmoid** squashes input into the range (0,1).  
+- The **gradient (derivative)** is maximum at `x = 0`, but becomes **very small** for large |x|.  
+- In deep networks, multiplying many small gradients leads to the **vanishing gradient problem**.  
+""")