I want to differentiate (1) wrt the vector $x \in \mathbb{R}^n$ where $w(x)$ is a zero-one diagonal indicator matrix
\begin{align} \frac{1}{2} g(x)^T w(x) g(x) &&&& (1) \end{align}
Since
\begin{align} \frac{1}{2} g(x)^T w(x) g(x)= \langle g(x),g(x) \rangle \end{align} Thus, \begin{align} \frac{d}{dx} \langle g(x),g(x) \rangle = \langle g(x)',g(x) \rangle + \langle g(x),g(x)' \rangle = 2 \langle g(x)',g(x) \rangle = (g(x)')^T w(x)g(x) \end{align}
but the notes I study from says the outcome is $g(x)'w(x)g(x)$
Why is that?