The Chain Rule

Business Calculus · Axiom Academy

Differentiating composite functions Consider the function h(x) = (x^2 + 1)^5 . This is a composite function —a function inside another function. We can write: h(x) = f(g(x)) where we substitute g(x) into f . If h(x) = f(g(x)) , then the derivative is: Derivative of outer × Derivative of inner In words: Differentiate the outer function (keeping the inner unchanged), then multiply by the derivative of the inner function. If y = f(u) and u = g(x) , then: The du's "cancel" like fractions! Find the derivative of h(x) = (x^2 + 1)^5 Here are the most common chain rule patterns you'll encounter: Example: Find the derivative of y = Click each problem to reveal the solution: Sometimes you need to apply the chain rule multiple times! For ((x^2+1)^3)^2 , work from outside in. Find the derivative of f(x) = (5x - 1)^3 You've mastered the Chain Rule! "Derivative of outer times derivative of inner" The chain rule is one of the most important differentiation techniques. Combined with the power, product, and quotient rules, you can now differentiate almost any function!

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