Chain Rule Calculator

What the chain rule does

Nested functions are everywhere in calculus: trig of a polynomial, exp of a product, roots of sums. The chain rule says (f(g(x)))′ = f′(g(x)) · g′(x). A chain rule calculator applies that pattern automatically and shows the labeled working so you can rehearse the same moves on exams.

Mentally name the outer and inner pieces before you press Calculate. If the step list matches your prediction, you are ready for closed-book problems.

How to use this chain rule derivative calculator

1. Enter a composition such as sin(x^2) or e^(3*x).
2. Press Calculate and find the chain-rule line in the steps.
3. Check that the outer derivative is evaluated at the inner function.
4. Confirm the inner derivative multiplies correctly after simplification.

Chain rule practice set

• sin(x^2) — cos(x²) · 2x
• e^(3*x) — e^(3x) · 3
• ln(x^2+1) — 1/(x²+1) · 2x
• sqrt(x^2+1) or (x^2+1)^(1/2) — power form + chain
• (2*x+1)^5 — power rule on the outside
• cos(3*x) — −sin(3x) · 3

Mix with other rules: x*sin(x^2) needs product and chain; e^(3*x)/(x+1) needs quotient and chain. The engine labels each piece.

Common mistakes the steps catch

Forgetting to multiply by the inner derivative, differentiating the inside as if it were the outside, and dropping the chain factor on e^(u) or ln(u) are the usual slips. Reading the free step list after your own attempt is faster than re-deriving from scratch.

For products or quotients without composition, switch to the product rule calculator or quotient rule calculator. For the general tool, use the derivative calculator.

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