Derivative of Dot Product of Vector-Valued Functions

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Theorem

Let $\mathbf a: \R \to \R^n$ and $\mathbf b: \R \to \R^n$ be differentiable vector-valued functions.


The derivative of their dot product is given by:

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


Proof 1

Let:

$\mathbf a: x \mapsto \tuple {\map {a_1} x, \map {a_2} x, \ldots, \map {a_n} x}$
$\mathbf b: x \mapsto \tuple {\map {b_1} x, \map {b_2} x, \ldots, \map {b_n} x}$


Then:

\(\ds \map {\frac \d {\d x} } {\mathbf a \cdot \mathbf b}\) \(=\) \(\ds \map {\frac \d {\d x} } {\sum_{i \mathop = 1}^n a_i b_i}\) Definition of Dot Product
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \map {\frac \d {\d x} } {a_i b_i}\) Sum Rule for Derivatives
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \paren {\map {\frac \d {\d x} } {a_i} b_i + a_i \map {\frac \d {\d x} } {b_i} }\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \map {\frac \d {\d x} } {a_i} b_i + \sum_{i \mathop = 1}^n a_i \map {\frac \d {\d x} } {b_i}\) Summation is Linear
\(\ds \) \(=\) \(\ds \dfrac {\d \mathbf a} {\d x} \cdot \mathbf b + \mathbf a \cdot \dfrac {\d \mathbf b} {\d x}\) Definition of Dot Product

$\blacksquare$


Proof 2

Let $\mathbf v = \mathbf a \cdot \mathbf b$.

Then:

\(\ds \dfrac {\d \mathbf v} {\d x}\) \(=\) \(\ds \lim_{h \mathop \to 0} \dfrac {\map {\mathbf v} {x + h} - \map {\mathbf v} x} h\) Definition of Derivative of Vector-Valued Function
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \dfrac {\map {\mathbf a} {x + h} \cdot \map {\mathbf b} {x + h} - \map {\mathbf a} x \cdot \map {\mathbf b} x} h\) Definition of $\mathbf v$
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \dfrac {\map {\mathbf a} {x + h} \cdot \map {\mathbf b} {x + h} - \map {\mathbf a} {x + h} \cdot \map {\mathbf b} x + \map {\mathbf a} {x + h} \cdot \map {\mathbf b} x - \map {\mathbf a} x \cdot \map {\mathbf b} x} h\) rearranging
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \map {\mathbf a} {x + h} \cdot \dfrac {\map {\mathbf b} {x + h} - \map {\mathbf b} x} h + \lim_{h \mathop \to 0} \dfrac {\map {\mathbf a} {x + h} - \map {\mathbf a} x} h \cdot \map {\mathbf b} x\) rearranging
\(\ds \) \(=\) \(\ds \mathbf a \cdot \dfrac {\d \mathbf b} {\d x} + \dfrac {\d \mathbf a} {\d x} \cdot \mathbf b\) Definition of Derivative of Vector-Valued Function

$\blacksquare$


Also see


Sources

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