Chain Rule Calculator
Chain rule calculator with steps. See the outer derivative, the inner derivative, and the product for composite functions such as sin(2x).
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<iframe src="https://derivativecalculatorai.com/chain-rule-calculator" title="Derivative Calculator AI" width="100%" height="640" style="border:0"></iframe>The chain rule differentiates a function of a function. If y = f(u) and u = g(x), then dy/dx = f'(u) · g'(x). For sin(2x) the outside is sin(u), whose derivative is cos(u), and the inside 2x has derivative 2. Multiply to get 2 cos(2x).
The factor from the inside is the part that gets dropped. cos(2x) is not the derivative of sin(2x). e^(2x) is not the derivative of itself; the 2 stays in front. ln(2x) is the exception that looks like a cancellation: the chain rule multiplies by 2 and the 2 in the denominator cancels, leaving 1/x.
Enter a composite such as sin(2*x), exp(x^2), or (x+1)^5. The calculator names the outer function, the inner function, both derivatives, and the simplified result. You can check a derivative you computed by hand on the same page.
Rule and examples
Frequently asked questions
What is the chain rule?
The derivative of f(g(x)) is f'(g(x)) times g'(x).
What is the derivative of sin(2x)?
2 cos(2x). The 2 is the derivative of the inside.
When is the inner derivative equal to 1?
When the inside is x itself, as in sin(x) or e^x. The chain rule still applies; the extra factor is invisible.
Does the chain rule replace the product rule?
No. A product of two functions of x uses the product rule. Each factor may then need the chain rule of its own.