def sigmoid_derivative(x):
    """Derivative of sigmoid function"""
    s = sigmoid(x)
    return s * (1 - s)  # Derivative formula: sigmoid(x) * (1 - sigmoid(x))

# Test derivative
test_vals = np.array([-2, -1, 0, 1, 2])
sigmoid_vals = sigmoid(test_vals)
derivative_vals = sigmoid_derivative(test_vals)

print("Sigmoid and its derivative:")
for i, val in enumerate(test_vals):
    print(f"  x={val:2.0f}: sigmoid={sigmoid_vals[i]:.3f}, derivative={derivative_vals[i]:.3f}")

# Visualize both functions
x_range = np.linspace(-6, 6, 100)
y_sigmoid = sigmoid(x_range)
y_derivative = sigmoid_derivative(x_range)

plt.figure(figsize=(10, 4))
plt.plot(x_range, y_sigmoid, 'b-', label='sigmoid(x)', linewidth=2)
plt.plot(x_range, y_derivative, 'r--', label="sigmoid'(x)", linewidth=2)
plt.title('Sigmoid Function and Its Derivative')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()