The chain rule is a means to take the derivative of a composite function (a function inside another function). The rule states that the rate of change of the composite function is equal to the derivative of the outer function times the derivative of the inner function.
For F(x) = f(g(x)), then F'(x) = f'(g(x)) * g'(x)
To use the chain rule, the following rules are required. [1]
- Such functions must be differentiable themselves
- The function must be a composite function of two or more functions
To do the chain rule. [1]
- Differentiate the outer function, keeping the inner function the same.
- Multiply this by the derivative of the inner function.
For F(x) = f(g(x)), then F'(x) = f'(g(x)) * g'(x)
Peeling the Onion
In the chain rule, the basic idea is to peel the onion from the outside. You want to take the derivative of a function within a function within a function. You take the derivative of the outermost function relative to the stuff that’s inside it, then multiply that by the derivative of the inside expression, relative to the expression inside the expression, and so on, all the way down to the tiniest little x all the way inside. (And some people even stick a “1” on at the end, because the derivative of an x is just 1. I think that’s overdoing it a bit.)
The Chain Rule needs quite a lot of imagination to see these formulas as expressions within expressions, and ideally you should have a friend sit by you and point out how to “peel the onion” layer by layer. [2]
Example
f(x) = x5
g(x) = 4x2 – 3
F(x) = f(g(x)) = (4x2 – 3)5
Find F‘(x) using the chain rule.
f‘(x) = 5x4
g‘(x) = 8x
∴ F‘(x) = 5(4x2 – 3)4 * 8x = 40x(4x2 – 3)4
References
[1] MATHS at Home. “The Chain Rule Made Easy: Examples and Solutions – mathsathome.com.” Accessed June 11, 2026. https://mathsathome.com/chain-rule-differentiation/.
[2] Peterson, Dave. “How to Think About the Chain Rule – The Math Doctors.” The Math Doctors, December 29, 2023. https://www.themathdoctors.org/how-to-think-about-the-chain-rule/.
Additional Reading
[ ℰ ] Calculus. “3.1 The Chain Rule.” Accessed June 11, 2026. https://calculus.flippedmath.com/31-the-chain-rule.html.
“Chain Rule And Composite Functions.” BYJU’S, December 19, 2015. https://byjus.com/maths/chain-rule/.
CUEMATH. “Chain Rule – Theorem, Proof, Examples.” Accessed June 11, 2026. https://www.cuemath.com/chain-rule-formula/.
Dawkins, Paul. “Calculus I.” Chain Rule, Paul’s Online Notes, April 23, 2026. https://tutorial.math.lamar.edu/classes/calci/chainrule.aspx.
Dawkins, Paul. “Calculus I.” Chain Rule (Practice Problems), Paul’s Online Notes, November 16, 2022. https://tutorial.math.lamar.edu/problems/calci/chainrule.aspx.
“Derivation of the Chain Rule.” www2.math.uconn.edu. Accessed June 11, 2026. https://www2.math.uconn.edu/~stein/virtual/Notes/Calculus/chainrule.pdf.
GeeksforGeeks. “Chain Rule: Theorem, Formula and Solved Examples.” GeeksforGeeks, March 27, 2022. https://www.geeksforgeeks.org/maths/chain-rule-formula/.
Strang, Gilbert & Edwin “Jed” Herman. “3.6: The Chain Rule.” LibreTexts, July 11, 2016. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/03%3A_Derivatives/3.06%3A_The_Chain_Rule.
Videos
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