In Python, we can compute derivatives symbolically using sympy and visualize them using matplotlib.

1. Computing Derivatives Symbolically

# Define symbolic variable x
x = sp.symbols('x')
# Define the functions
f1 = sp.exp(x)  
f2 = 1 / (1 + sp.exp(-x))  
# Compute derivatives symbolically
df1 = sp.diff(f1, x)  
df2 = sp.diff(f2, x)

Explanation:

• We define x as a symbolic variable using sp.symbols('x');
• The function sp.diff(f, x) computes the derivative of f with respect to x;
• This allows us to manipulate derivatives algebraically in Python.

2. Evaluating and Plotting Functions and Their Derivatives

# Convert symbolic functions to numerical functions for plotting
f1_lambda = sp.lambdify(x, f1, 'numpy')
df1_lambda = sp.lambdify(x, df1, 'numpy')
f2_lambda = sp.lambdify(x, f2, 'numpy')
df2_lambda = sp.lambdify(x, df2, 'numpy')

Explanation:

• sp.lambdify(x, f, 'numpy') converts a symbolic function into a numerical function that can be evaluated using numpy;
• This is required because matplotlib and numpy operate on numerical arrays, not symbolic expressions.

3. Printing Derivative Evaluations for Key Points

To verify our calculations, we print derivative values at x = [-5, 0, 5].

# Evaluate derivatives at key points
test_points = [-5, 0, 5]
for x_val in test_points:
    print(f"x = {x_val}: e^x = {f2_lambda(x_val):.4f}, e^x' = {df2_lambda(x_val):.4f}")
    print(f"x = {x_val}: sigmoid(x) = {f4_lambda(x_val):.4f}, sigmoid'(x) = {df4_lambda(x_val):.4f}")
    print("-" * 50)