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)
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