Dot product of a vector and del vector identity

Question

How did this identity come about.

$V \cdot \nabla V=1/2 \nabla (V^2)$

Answer 1

Suppose $\;V=(f_1,...,f_n)\;$ , so that

$$\nabla V=\left(\frac{\partial f_1}{\partial x_1},\,\ldots,\frac{\partial f_n}{\partial x_n}\right)\implies V\cdot\nabla V=\sum_{k=1}^nf_k\frac{\partial f_k}{\partial x_k}$$

and now, assuming $\;V^2=V\cdot V=\sum\limits_{k=1}^nf_k^2$ , we have

$$\frac12\nabla V^2=\frac12\sum_{k=1}^n\frac{2f_k\partial f_k}{\partial x_k}$$ Now just compare both expressions above...