Derivative of Composite Function/3 Functions

Theorem

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

\(\ds y\)

\(=\)

\(\ds \map f u\)

\(\ds u\)

\(=\)

\(\ds \map g v\)

\(\ds h\)

\(=\)

\(\ds \map h x\)

Then:

$\dfrac {\d y} {\d x} = \dfrac {\d y} {\d u} \cdot \dfrac {\d u} {\d v} \cdot \dfrac {\d v} {\d x}$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chain rule (for differentiation)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chain rule (for differentiation)