1def numerical_derivative(f, x, h=1e-5):
2    """Calculate derivative using central difference method"""
3    return (f(x + h) - f(x - h)) / (2 * h)
4
5# Example: derivative of x^2 at x=3
6result = numerical_derivative(lambda x: x**2, 3)
7print(f"Derivative at x=3: {result}")  # Should be ~6

1from sympy import symbols, diff, sin, cos, exp
2
3x = symbols('x')
4# Define function
5f = x**3 + 2*x**2 - 5*x + 1
6# Calculate derivative
7f_prime = diff(f, x)
8print(f"f'(x) = {f_prime}")  # 3x^2 + 4x - 5

1def simpsons_rule(f, a, b, n=100):
2    """Approximate integral using Simpson's Rule"""
3    if n % 2 == 1:
4        n += 1  # Must be even
5    h = (b - a) / n
6    x = [a + i*h for i in range(n+1)]
7    y = [f(xi) for xi in x]
8    
9    integral = y[0] + y[-1]
10    integral += 4 * sum(y[i] for i in range(1, n, 2))
11    integral += 2 * sum(y[i] for i in range(2, n-1, 2))
12    integral *= h / 3
13    
14    return integral

1import numpy as np
2
3def gradient_descent(f, grad_f, x0, learning_rate=0.01, iterations=1000):
4    """Minimize function using gradient descent"""
5    x = x0
6    history = [x]
7    
8    for i in range(iterations):
9        gradient = grad_f(x)
10        x = x - learning_rate * gradient
11        history.append(x)
12    
13    return x, history
14
15# Example: minimize f(x) = x^2
16result, path = gradient_descent(
17    lambda x: x**2,
18    lambda x: 2*x,
19    x0=5.0
20)
21print(f"Minimum found at x = {result}")