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Let's consider the well known "fake" proof below for the derivative of the composition of functions:

Let $E,G$ be intervals of $\mathbb{R}$, let $F$ a subset of a normed vector space, let $f:E\rightarrow F,g:G\rightarrow \mathbb{R}$ be derivable functions on $E,G$. We suppose that $g(G)\subset E$.

Let $a\in G$, we prove that $(f\circ g)'(a)=g'(a)\cdot f'(g(a))$ :

$\displaystyle\dfrac{f\circ g(x)-f\circ g(a)}{x-a}=\dfrac{f\circ g(x)-f\circ g(a)}{g(x)-g(a)}\cdot\dfrac{g(x)-g(a)}{x-a}\to_{x\to a}(f\circ g)'(a)=g'(a)\cdot f'(g(a))$


This proof might be wrong because we can't assume that $g(x)-g(a)$ isn't $0$ n a neighborhood of $a$, but it nonetheless seems really nice, and the "real" proof I know isn't quite as nice.

What's the "nicest" proof of this property that doesn't use "too high level" maths (understand : a proof on the same level as the one above) ?

PS : I know that "nice" isn't well defined to describe proofs, but I'm sure you will get the idea.

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