Chain rule derivative calculator
The Chain rule derivative calculator computes chain rule derivative from the relation (f(g))' = f'(g) g'. It takes 2 inputs (f(x) (composite, e.g. sin(x^2)), evaluate at x) and returns the derivative at x. These tools are for students working through a calculus course and for anyone in a quantitative subject or job who needs one step worked out for the function or values in front of them: a derivative, a limit, a definite or indefinite integral, a series expansion, or a root estimate. Enter your values below and the result updates instantly, and you can share a permalink that pre-fills the exact calculation. Enter the function, coefficients or interval endpoints in the form the tool asks for, stay consistent about which variable the result is taken with respect to, and treat a decimal output as a rounded reading of a value that may have an exact closed form, so carry the extra digits into any later step and round only at the end. For example, with f(x) (composite, e.g. sin(x^2)) = exp(2*x), evaluate at x = 0, the derivative at x works out to 2, and the worked example further down the page shows every step so you can follow the arithmetic and reproduce it by hand. This is a standard formula rather than a jurisdiction-specific rule, so it does not depend on an external rate or schedule, and the figure above each result carries the date it was last verified. A result means what you expect only where the function is defined and well behaved over the point or interval you give it: a discontinuity, a vertical asymptote, a domain edge, or a series that does not converge can still return a plausible looking number, and any step based or numerical method returns an approximation whose closeness depends on the step size or the number of terms rather than an exact value.
With f(x) (composite, e.g. sin(x^2)) = exp(2*x), Evaluate at x = 0, the result is 2.
Applies to: any numeric inputs. Method: a standard formula, not a jurisdiction-specific rule. How we document and test our methods, checked 2026-07-09.
The formula
(f(g))' = f'(g) g'
Worked example
With f(x) (composite, e.g. sin(x^2)) = exp(2*x), Evaluate at x = 0:
- differentiate via chain rule
- evaluate at x
- Derivative at x = 2
This worked example is one of the automated golden-value tests this calculator must pass before it can publish.
What this assumes
- Inputs are real numbers in the units shown.
- The result is the exact value of (f(g))' = f'(g) g'; general information, not professional advice.
Frequently asked questions
What formula does this use?
(f(g))' = f'(g) g'. This is a standard formula rather than a jurisdiction-specific rule.
Does the result ever change over time?
No. This is a pure formula with no external rate, so the same inputs always give the same result.
Official sources and verification
- Method: a standard formula rather than a jurisdiction-specific rule; how we document and test our methods, checked 2026-07-09.
Reviewed by the CalculatorHub team, edited by James Graham, 2026-07-09. See our methodology. General information, not professional advice.