What is the chain rule?

The chain rule allows us to differentiate composite functions — functions inside functions.

📅 1. May 2026 👁️ 1 views 📂 kalkulus 🇳🇴 Les på norsk

Many functions are composed of two or more functions. To differentiate these we use the chain rule.

[f(g(x))]' = f'(g(x)) · g'(x)

We differentiate the outer function and multiply by the derivative of the inner function.

Method:
1. Identify the outer function f and inner function g
2. Differentiate the outer: f'(g(x))
3. Differentiate the inner: g'(x)
4. Multiply the results together

Examples

Example 1: Differentiate h(x) = (x² + 3)⁵

Outer: f(u) = u⁵ → f'(u) = 5u⁴
Inner: g(x) = x² + 3 → g'(x) = 2x
h'(x) = 5(x² + 3)⁴ · 2x = 10x(x² + 3)⁴

Example 2: Differentiate h(x) = sin(3x)

Outer: f(u) = sin(u) → f'(u) = cos(u)
Inner: g(x) = 3x → g'(x) = 3
h'(x) = cos(3x) · 3 = 3cos(3x)

Common mistake: Many forget to multiply by the derivative of the inner function. The chain rule always requires this extra factor!

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.