Maximum rate of increase- Directional derivatives

Question

If I have been asked:

Find the directional derivative of $$f(x, y, z) = \frac{x}{y}−\frac{y}{z}$$ at the point $P(0, −1, 2)$ in the direction from $P$ to $Q(3, 1, −4)$ and find the maximum rate of increase.

For the second part 'max rate of...' is that just calculating the maximum directional derivative?

Thanks

Answer 1

Recall that the maximum rate of increase is given by the norm of the gradient vector $|\nabla f(P)|$ indeed the directional derivative in direction $\vec u$ is given by

$$\nabla f(P)\cdot \vec u=|\nabla f(P)|\cos \theta$$