# Derivative of Dot Product of Vector-Valued Functions

## 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

$\blacksquare$