Math · Calculus · Concept
Rates through a composition
A change in x passes through an inner function before reaching the outer one.
Follow the intermediate quantity
Write u=g(x) and y=f(u). A small input change makes Δu approximately g′(x)Δx. Near u, the outer output changes approximately f′(u)Δu. Combining the local linear approximations multiplies their rates.
Evaluate the outer rate at the inner output
The chain rule uses the slope of f at g(x), not at x. A rigorous derivative statement follows from the error terms in those linear approximations, including when g′ is zero; literal cancellation of differentials is only a mnemonic.
Track domains and units
Require g to be differentiable at x and f differentiable at g(x). Output-per-intermediate units multiplied by intermediate-per-input units give output-per-input units. Trigonometric differentiation uses radians.
Common mistakes
- Evaluating f′ at x instead of g(x) changes the rule.
- Forgetting the inner rate is especially easy with affine inner functions.
Work through an example
Follow the inner and outer rates.
Differentiate (3x+1)² →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.