Chains Within Chains

Calculus 2 · Axiom Academy

A composite function is built in layers. The chain rule wraps those layers up into a derivative — and u-substitution is how you peel them back apart to integrate. Functions hidden inside functions Look at . It isn't one thing — it's three functions chained one after the next, like an assembly line. To differentiate it you work outside-in with the chain rule; to integrate something built this way you go inside-out with a substitution. Both start by seeing the layers, so let's watch the function get built first. Watch a value run through the machine. It enters as and flows through three stages: the innermost core goes first, that result is fed to the middle stage , and that feeds the outer stage . Each stage wraps one more layer onto the running expression — built inside-out, the order you'd actually evaluate it. The innermost stage runs first, its result is squared, then fed to sine — that nesting is exactly what the chain rule and u-substitution navigate. Going backward — integrating — starts with a hunt. In , slide through the candidate inner functions. For each guess at , the panel shows the derivative you'd need, and checks whether that derivative is already sitting in the integrand. When the match lights up, that's your substitution. The right inner function is the one whose derivative is already in the integrand — here, choosing u = x² puts 2x dx exactly where you need it. Peeling the layers: the substitution

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