Proposition: Chain Rule

Let $D$ and $E$ be subsets of the real numbers $\mathbb R$ and let $f:D\to\mathbb R,$ $g:E\to\mathbb R,$ be functions such that $f(D)\subset E.$ Furthermore, let the function $f$ be differentiable at a point $x\in D$ and let $g$ be differentiable at the point $y:=f(x)\in E.$ Then the composition of functions $$g\circ f:D\to\mathbb R$$ is differentiable at $x$ and its derivative is given by $$(g\circ f)'(x)=g'(f(x))f'(x).$$ This rule also known as the chain rule.

Proofs: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983