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:
$$f(x,y) = \sqrt{\frac{xy}{x+y}}$$
Best Answer
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.