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
- Derivative of Vector Cross Product of Vector-Valued Functions
- Derivative of Product of Real Function and Vector-Valued Function
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Formulas involving Derivatives: $22.23$