Reversing the Chain Rule

Calculus 1 · Axiom Academy

Every chain-rule derivative can be run backwards. That reverse gear is u‑substitution. The chain rule, played in reverse When you differentiate a composition like , the chain rule leaves a fingerprint: the answer is a product , and one factor is the derivative of the inside. That fingerprint is the whole reason hard-looking integrals can suddenly become easy — if you can spot it, you can run the chain rule backwards. Watch the chain rule fire forward on : out comes , with that extra 2x — the derivative of the inside — tagged. Then the arrow flips: integrating that same expression walks the steps in reverse, straight back to . Differentiate and that 2x appears; integrate and it gets absorbed. u‑substitution is just this reverse trip, organized. The reverse trick only works when the integrand really is a chain-rule leftover — some inner function together with its own derivative. Tap each integral to check it: a good candidate lights up its inside g(x) and the matching g'(x) ; a near-miss tells you what s missing. Tap an integral to test it for the chain-rule pattern. When you see sitting under the integral sign, the substitution u=g(x) is waiting for you. Here is a candidate you just confirmed: . The art of u‑substitution is choosing u to be the inside . Make that choice below and watch each step of the reverse trip light up — the 2x gets swallowed by du , and a scary integral turns into one you already know.

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