The Chain Rule

Calculus 1 · Axiom Academy

Differentiate a function inside a function by multiplying the outer rate by the inner rate. 1. A Function Inside a Function Function composition feeds the output of one function straight into another. We write , read as "f of g of x." The inner function g runs first; the outer function f acts on what comes out. Take the composite . To evaluate it at x = 2 , push the number down the assembly line: first square it, then take the sine of the result. Here is the key idea. A small nudge in the input gets amplified twice on its way through. The inner function stretches it by g'(x) ; the outer function then stretches that by f'(g(x)) . Two stretches in a row means the factors multiply . The working rule is short: differentiate the outside, leaving the inside alone, then multiply by the derivative of the inside. Apply it to . Identify: outer , inner g(x) = x^2 Outside at the inside, kept whole: Multiply by the inner derivative: You've seen the Chain Rule as an assembly line of rates: differentiate the outside, keep the inside, and multiply by the inside's derivative. Scroll up to revisit any step.

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