$\frac{\partial}{\partial a}\frac{1}{2}\|\mathbf{f}(a)\|_2^2$ and $\mathbf{f}$ is $\mathbb R^1\to\mathbb R^n$.
The problem I encountered is that the chain rule dimension. Is the following derivation correct?
$$\frac{\partial}{\partial a}\frac{1}{2}\|\mathbf{f}(a)\|_2^2=\mathbf{f}(a)\cdot\frac{\partial}{\partial a}\mathbf{f}(a)$$ However I do not know how does the $\cdot$ operate.