Derivatives are fundamental in calculus and widely used in data science, optimization, and machine learning. 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)