Proof of the derivative of the compostion of functions.

Question

This question comes from the proof of the derivative of a composite function which I lifted straight from Proof Wiki

https://proofwiki.org/wiki/Derivative_of_Composite_Function:

Let $f, g, h$ be continuous real functions such that: $\forall x \in \mathbb R: h \left({x}\right) = f \circ g \left({x}\right) = f \left({g \left({x}\right)}\right)$

Then: $h' \left({x}\right) = f' \left({g \left({x}\right)}\right) g' \left({x}\right)$

where $h'$ denotes the derivative of $h$.

Proof:

Let $g \left({x}\right) = y$, and let

$g(x+δx) = y+δy$

Thus:

Thus:

$δy→0\ \ \ \ \ $ as $\ \ \ \ \ δx→0$

and:

$\frac{δy}{δx}→g′(x) \ \ \ \ \ \ \ \ \ (1)$

Case 1:

Suppose $g′(x)≠0$ and that $δx$ is small but non-zero.

Then $δy≠0$ from (1) above, and:

$lim_{δx→0} \frac{h(x+δx)−h(x)}{δx} = lim_{δx→0}\frac{f(g(x+δx))−f(g(x))}{g(x+δx)−g(x)}$ $\frac{g(x+δx)−g(x)}{δx}$
= $lim_{δx→0}\frac{f(y+δy)−f(y)}{δy}$ $\frac{δy}{δx}$
= $f′(y)g′(x)$

My question: Why does the assumption that $g'(x)$ $\neq$ $0$ imply that $dy$ $\neq$ $0$ and why does this in turn imply that the expression $\ \ \ \ \ lim_{δx→0} \frac{h(x+δx)−h(x)}{δx}\ \ \ \ \ $

is equal to $\ \ \ \ \ \ \ \ \ \ \ \ \ \ lim_{δx→0}\frac{f(g(x+δx))−f(g(x))}{g(x+δx)−g(x)}\frac{g(x+δx)−g(x)}{δx}$

Answer 1

If $g'(x)$ $\neq$ $0$

Since $g'(x) = \frac{dy}{dx}$, and we have also assumed $dx$ is small but non-zero. So, the product of two non-zero values yield non-zero result. implies that $dy$ $\neq$ $0$

Since $dy$ $\neq$ $0$, by our definition of $dy$, we know this is equivalent to saying ${g(x+δx)−g(x)}$$\neq$ $0$. Then, this means it's valid to multiply top and bottom of the fraction by ${g(x+δx)−g(x)}$. Which in turn yields:

The expression$\ \ \ \ \ lim_{δx→0} \frac{h(x+δx)−h(x)}{δx}\ \ \ \ \ $

is equal to $\ \ \ \ \ \ \ \ \ \ \ \ \ \ lim_{δx→0}\frac{f(g(x+δx))−f(g(x))}{g(x+δx)−g(x)}\frac{g(x+δx)−g(x)}{δx}$

Answer 2

Suppose $g'(x) > 0$. Then $\displaystyle \lim_{\delta x \to 0} \frac{g(x + \delta x) - g(x)}{\delta x} > 0$ and so there exists $\delta > 0$ with the property that $$0 < |\delta x| < \delta \implies \frac{g(x + \delta x) - g(x)}{\delta x} > 0 \implies g(x + \delta x) - g(x) \not= 0.$$ The last expression is just $\delta y$, so as long as $\delta x$ is sufficiently small you find $\delta y \not = 0$.

The same conclusion holds if $g'(x) < 0$.

The last line is just an algebraic manipulation: the expression $\dfrac{f(g(x + \delta x)) - f(g(x))}{\delta x}$ is being multiplied by $\dfrac{\delta y}{\delta y}$ and rearranged. As long as $\delta y \not= 0$ you aren't inadvertently multiplying the fraction by $\dfrac 00$.