prove derivative of dot product of A(u) and B(u) with respect to u.

Question

Suppose $\vec{A}$ and $\vec{B}$ are differentiable functions of a scalar u.
Prove:

$\Large \frac{d}{du}(\vec{A} \cdot \vec{B}) = \vec{A} \cdot \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} \cdot \vec{B}$

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ok so i start with definition of derivative:

$\Large \frac{df(x)}{dx} = \lim \limits_{\Delta x \to \infty} \frac{f(x+\Delta x) - f(x)}{\Delta x}$

and apply it to the dot product of $\vec{A}$ and $\vec{B}$:

$\Large \frac{d}{du} (\vec{A} \cdot \vec{B}) = \lim \limits_{\Delta u \to \infty} \frac{(\vec{A}+\Delta\vec{A}) \cdot (\vec{B}+\Delta\vec{B}) -\vec{A}\cdot\vec{B}}{\Delta u}$

appying "foil" method to expand dot produt in numerator:

$\Large \frac{d}{du} (\vec{A} \cdot \vec{B}) = \lim \limits_{\Delta u \to \infty} \frac{\vec{A}\cdot\Delta B ~+~ \Delta \vec{A} \cdot \vec{B} ~+~ \Delta A \cdot \Delta B}{\Delta u}$

$\Large \frac{d}{du} (\vec{A} \cdot \vec{B}) = \lim \limits_{\Delta u \to \infty} \frac{\vec{A}\cdot\Delta B}{\Delta u} + \lim \limits_{\Delta u \to \infty} \frac{\Delta \vec{A} \cdot \vec{B}}{\Delta u} + \lim \limits_{\Delta u \to \infty} \frac{\Delta A \cdot \Delta B}{\Delta u}$

$\Large \frac{d}{du} (\vec{A} \cdot \vec{B}) = \vec{A} \cdot \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} \cdot \vec{B} + \lim \limits_{\Delta u \to \infty} \frac{\Delta A \cdot \Delta B}{\Delta u}$

now for the part that I don't understand..why does this piece goto zero:

$\Large \lim \limits_{\Delta u \to \infty} \frac{\Delta A \cdot \Delta B}{\Delta u} = 0?$

Answer 1

I assume we are working in $\mathbb{R}^n$, take $\{e_i\}$ the standard orthonormal basis.

Then, $\mathbf{A}(u) = \sum a_i(u) e_i$, $\mathbf{B}(u) = \sum b_i(u) e_i$, and $(\mathbf{A}\cdot\mathbf{B})(u) = \sum a_i(u) b_i(u)$. Now evaluating the derivative is much easier because we only have to evaluate each term in the sum. We do that with the product rule:

\begin{equation} \frac{d}{du}(\mathbf{A}\cdot\mathbf{B} )(u) = \sum a_i(\frac{d}{du}b_i) + \sum b_i(\frac{d}{du}a_i) = \mathbf{A}\cdot\frac{d\mathbf{B}}{du} + \frac{d\mathbf{A}}{du} \cdot\mathbf{B}. \end{equation}

Answer 2

Multiplying top and bottom by $\Delta u$, we get $$\begin{align} \lim_{\Delta u \rightarrow 0}\frac {\Delta\vec A\cdot\Delta\vec B}{\Delta u} &=\lim_{\Delta u \rightarrow 0}\frac {\Delta\vec A}{\Delta u} \cdot\frac{\Delta\vec B}{\Delta u}~\Delta u \\[1ex]&= \frac {d\vec A}{du}\cdot \frac {d\vec B}{du}\lim_{\Delta u \rightarrow 0} \Delta u\\[1ex] &= 0\end{align}$$