Chain rule
If y = f(g(x)), dy/dx = f'(g(x)) × g'(x). For composite functions.
Y13 differentiation. Apply when one function is ‘wrapped’ in another. e.g. y = (3x + 1)5: outer is □5, inner is 3x+1.
Worked examples
y = (3x + 1)5: y' = 5(3x+1)4 × 3 = 15(3x+1)4.
y = ex²: y' = ex² × 2x.
y = sin(2x): y' = cos(2x) × 2 = 2 cos 2x.
Frequently asked questions
How do I spot the ‘outer’ vs ‘inner’?
Outer is the last operation done; inner is what's inside it. Differentiate outer keeping inner intact, multiply by derivative of inner.
Multiple chains?
Apply repeatedly: y = esin x² = e^(sin(x²)) → derivative is e^(sin x²) × cos(x²) × 2x.