← Unit 3: Differentiation — Composite, Implicit & Inverse Functions
The Chain Rule
The chain rule differentiates a composite function f(g(x)) by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.
Outer times inner
If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). In words: differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside.
You'll almost always combine this with the product/quotient rules once functions get complicated — differentiate from the outside in, one layer at a time.
Worked Example
Differentiate h(x) = sin(x²).
- Outer function: sin(u); inner function: u = x².
- Derivative of outer (leaving inner alone): cos(x²).
- Derivative of inner: d/dx[x²] = 2x.
- Multiply: h'(x) = cos(x²) · 2x = 2x·cos(x²).
Answer: h'(x) = 2x·cos(x²)
Key terms (1)
- Chain rule
- d/dx[f(g(x))] = f'(g(x))·g'(x)
Practice Quiz
Question 1 of 3Score so far: 0/0
Differentiate f(x) = (3x+1)⁵.