Directional derivative of a scalar field in the direction of fastest increase of another such field Suppose $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are scalar fields. What expression represents the directional derivative of $f$ in the direction in which $g$ is increasing the fastest?

The vector field encoding the greatest increase in $g$ is the gradient of $g$, so the directional derivative of $f$ in the direction of $\text{grad}(g)$ is just $\text{grad}(f)\bullet \text{grad}(g)$. 
