# Derivative of Composite Function/Corollary

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

## Proof

\(\displaystyle \frac {\d y} {\d x} \frac {\d x} {\d u}\) | \(=\) | \(\displaystyle \frac {\d y} {\d u}\) | Derivative of Composite Function | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \frac {\d y} {\d x}\) | \(=\) | \(\displaystyle \frac {\paren {\dfrac {\d y} {\d u} } } {\paren {\dfrac {\d x}{\d u} } }\) | divide both sides by $\dfrac {\d x} {\d u}$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 13$: General Rules of Differentiation: $13.13$