Finding the rate of change in direction

Question

How would I go about answering this question? Would anyone be able to point me in the right direction?

Find a unit vector in the direction in which $f$ increases most rapidly at $P$, and find the rate of change of $f$ at $P$ in that direction: $$f(x,y) = \sqrt{\frac{xy}{x+y}}$$

Answer 1

I suppose we should point you in the direction of maximal increase?

At any rate, you're looking for the gradient of $f$; for any differentiable function $f$, $\nabla f$ (or grad$(f)$ if you prefer) points in the direction of maximal increase at any given point.

Answer 2

Compute $\nabla f = \left({\partial f\over \partial x}, {\partial f\over\partial y}\right)$. This is a vector that points in the direction of fastest increase of $f$. Evaluate it at point $P$. Then normalize the vector to unit length (because they asked for a unit vector). The rate of change of $f$ in that direction is the dot product of $\nabla f$ with the unit vector.