Derivative of Scalar Triple Product of Vector-Valued Functions

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Theorem

Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be differentiable vector-valued functions in Cartesian $3$-space.


The derivative of their scalar triple product is given by:

$\map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} } = \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c} + \mathbf a \cdot \paren {\mathbf b \times \dfrac {\d \mathbf c} {\d x} }$


Proof

\(\ds \map {\dfrac \d {\d x} } {\mathbf a \cdot \paren {\mathbf b \times \mathbf c} }\) \(=\) \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \map {\dfrac \d {\d x} } {\mathbf b \times \mathbf c}\) Derivative of Dot Product of Vector-Valued Functions
\(\ds \) \(=\) \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c + \mathbf b \times \dfrac {\d \mathbf c} {\d x} }\) Derivative of Vector Cross Product of Vector-Valued Functions
\(\ds \) \(=\) \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \paren {\mathbf b \times \mathbf c} + \mathbf a \cdot \paren {\dfrac {\d \mathbf b} {\d x} \times \mathbf c} + \mathbf a \cdot \paren {\mathbf b \times \dfrac {\d \mathbf c} {\d x} }\) Dot Product Distributes over Addition

$\blacksquare$


Also see


Sources