Short answer
For a composite function y = f(g(x)), the derivative is y' = f'(g(x))*g'(x). Differentiate the outer function first, keep the inner expression unchanged, then multiply by the inner derivative.
- Identify inner and outer functions before any derivative step.
- Do not replace inner expression too early.
- Multiply by the inner derivative every time.
Decompose nested structure
Expressions like (3x-1)^5, sin(x^2), and e^(4x+7) are composite. They are not handled by a single power or trig rule alone.
Label inner function g(x) and outer function f(u). This makes the chain step explicit and easier to check.
Apply chain rule with clean substitution
Differentiate outer with respect to inner variable, then multiply by derivative of inner with respect to x.
For multiple nesting layers, repeat chain rule from outside to inside.
- Outer derivative keeps inner expression in place.
- Inner derivative usually introduces constants.
- Final check: dimensions and sign should match function behavior.
Differentiate a nested power
Find d/dx of (2x^3 - 5x)^4.
- Outer function is u^4, inner function is u = 2x^3 - 5x.
- Outer derivative gives 4u^3, so write 4(2x^3 - 5x)^3.
- Multiply by inner derivative d/dx(2x^3 - 5x) = 6x^2 - 5.
Result: Derivative is 4(2x^3 - 5x)^3(6x^2 - 5).
Common mistakes
- Applying power rule but forgetting inner derivative.
- Differentiating inner first and losing the outer structure.
- Mixing up which part is inner in trig or exponential composites.
- Expanding huge powers unnecessarily before differentiating.
Try one
What extra factor appears when differentiating cos(7x)?
A factor of 7 from the inner derivative, giving -sin(7x)*7.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.