Derivative of Composite Function/Corollary

Theorem

Let $f, g, h$ be continuous real functions such that:

$y = \map f u, x = \map g u$

Then:

$\dfrac {\d y} {\d x} = \dfrac {\paren {\dfrac {\d y} {\d u} } } {\paren {\dfrac {\d x} {\d u} } }$

for $\dfrac {\d x} {\d u} \ne 0$.

$\ds \frac {\d y} {\d x} \frac {\d x} {\d u}$

$=$

$\ds \frac {\d y} {\d u}$

$\ds \leadsto \ \ $

$\ds \frac {\d y} {\d x}$

$=$

$\ds \frac {\paren {\dfrac {\d y} {\d u} } } {\paren {\dfrac {\d x}{\d u} } }$

divide both sides by $\dfrac {\d x} {\d u}$

$\blacksquare$