What's the derivative ${\partial \over \partial x} \langle x, f(x)\rangle$?
According to the product rule it should be $1\cdot f(x) + x \cdot f'(x) $ but in my previous post I was told that this makes no sense.
Here $f: \mathbb R \to \mathbb R^2$ and $1$ is the constant one vector and $x \in \mathbb R^2$.
Best Answer
Let $f,g: I\subset \Bbb R \to \Bbb R^n$ be smooth maps, and $\langle \cdot, \cdot\rangle$ be the usual dot product in $\Bbb R^n$. So: $$\begin{align}\frac{\rm d}{{\rm d}x}\langle f(x),g(x)\rangle &= \frac{\rm d}{{\rm d}x}\sum_{i=1}^n f_i(x)g_i(x) \\ &= \sum_{i=1}^n \frac{\rm d}{{\rm d}x}(f_i(x)g_i(x)) \\ &= \sum_{i=1}^n(f'_i(x)g_i(x)+f_i(x)g'_i(x)) \\ &= \sum_{i=1}^nf'_i(x)g_i(x)+\sum_{i=1}^nf_i(x)g'_i(x)\\ &= \langle f'(x),g(x)\rangle + \langle f(x),g'(x)\rangle.\end{align}$$