Product Rule for Derivatives/General Result/3 Factors

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Theorem

Let $\map u x$, $\map v x$ and $\map w x$ be real functions differentiable on the open interval $I$.

Then:

$\forall x \in I: \map {\dfrac \d {\d x} } {u v w} = u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} + v w \dfrac {\d u} {\d x}$


Proof

Let $y = u v w$.

Then:

\(\ds y\) \(=\) \(\ds u \paren {v w}\)
\(\ds \) \(=\) \(\ds \dfrac {\d u} {\d x} \paren {v w} + u \map {\dfrac \d {\d x} } {v w}\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds v w \dfrac {\d u} {\d x} + u \paren {w \dfrac {\d v} {\d x} + v \dfrac {\d w} {\d x} }\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} + v w \dfrac {\d u} {\d x}\) simplification

$\blacksquare$


Sources

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