Calculus Fundamentals

import numpy as np

print("=" * 55)
print("DERIVATIVES IN ML")
print("=" * 55)

# Numerical derivative
def numerical_derivative(f, x, h=1e-7):
    return (f(x + h) - f(x - h)) / (2 * h)

# Power Rule: d/dx[x³] = 3x²
f = lambda x: x**3
x = 2.0
analytical = 3 * x**2
numerical = numerical_derivative(f, x)
print(f"\nf(x) = x³ at x = {x}")
print(f"  Analytical: f'(x) = 3x² = {analytical}")
print(f"  Numerical:  f'(x) ≈ {numerical:.6f}")

# Sigmoid Derivative
def sigmoid(x):
    return 1 / (1 + np.exp(-x))

def sigmoid_derivative(x):
    s = sigmoid(x)
    return s * (1 - s)

print("\n" + "=" * 55)
print("SIGMOID DERIVATIVE: σ'(x) = σ(x)(1 - σ(x))")
print("=" * 55)
for x in [-3, -1, 0, 1, 3]:
    s = sigmoid(x)
    sd = sigmoid_derivative(x)
    print(f"  x={x:2}: σ(x)={s:.4f}, σ'(x)={sd:.4f}")

print("\n💡 Notice: σ'(0) = 0.25 is maximum!")
print("   At |x| > 3, gradient nearly vanishes → 'vanishing gradient problem'")

# Chain Rule: d/dx[eˣ²] = 2x·eˣ²
print("\n" + "=" * 55)
print("CHAIN RULE: BACKPROPAGATION PREVIEW")
print("=" * 55)

# Loss = (y - σ(wx + b))²
w, x, b, y = 0.5, 2.0, 0.1, 1.0
z = w * x + b
a = sigmoid(z)
loss = (y - a) ** 2

# Backprop chain
dL_da = -2 * (y - a)
da_dz = sigmoid_derivative(z)
dz_dw = x

dL_dw = dL_da * da_dz * dz_dw

print(f"  Forward: z = wx+b = {z}")
print(f"           a = σ(z) = {a:.4f}")
print(f"           L = (y-a)² = {loss:.4f}")
print(f"\n  Backward (Chain Rule):")
print(f"    ∂L/∂a = {dL_da:.4f}")
print(f"    ∂a/∂z = {da_dz:.4f}")
print(f"    ∂z/∂w = {dz_dw}")
print(f"    ∂L/∂w = {dL_da:.4f} × {da_dz:.4f} × {dz_dw} = {dL_dw:.6f}")