Rules for derivatives

Computing limits every time would be slow. A few rules give the derivative of almost any formula.

Common derivatives

Combining functions

• Sum: (f + g)′ = f′ + g′. Derivatives work term by term, so 3x² + 5x has derivative 6x + 5.
• Product: (f·g)′ = f′·g + f·g′.
• Chain (a function inside a function): the derivative of f(g(x)) is f′(g(x)) · g′(x). The derivative of sin(x²) is cos(x²) · 2x.

Check it with a computer

The central difference (f(x + h) − f(x − h)) ÷ 2h is a much better estimate than the one-sided quotient. The example compares it with each rule at x = 1.

from math import sin, cos

def numeric(f, x, h=1e-5):
    return (f(x + h) - f(x - h)) / (2 * h)

x = 1.0
print("x^3:", round(numeric(lambda t: t ** 3, x), 4), " rule 3x^2:", 3 * x ** 2)
print("x sin x:", round(numeric(lambda t: t * sin(t), x), 4), " product rule:", round(sin(x) + x * cos(x), 4))
print("sin(x^2):", round(numeric(lambda t: sin(t * t), x), 4), " chain rule:", round(cos(x * x) * 2 * x, 4))

The numbers agree to four decimals, which is good evidence the rules were applied correctly. Automatic differentiation, used to train every neural network, applies the chain rule like this to millions of operations.

Common mistakes

• Forgetting the inner derivative in the chain rule: the derivative of sin(3x) is 3cos(3x), not cos(3x).
• Writing (f·g)′ = f′·g′. Try f = g = x: (x²)′ = 2x, not 1.

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