Calculus question, $f(x,y)=\frac{xy}{x+y}$

Question

Let $f(x,y)=\frac{xy}{x+y}$. Find a vector $u$ for which $D_u f(3,4)=0$.

I found the gradient of the function $f(x,y)$. Is the next step solving the gradient of $f(3,4)$ multiplied by $u = 0$?

Answer 1

Let u = (m,n) and grad(f) = (df/dx, df/dy) = (y^2/(x+y)^2, x^2/(x+y)^2). So Duf(3,4) = 0 <==>

u*grad(f)(3,4) = 0 <==> (16/49, 9/49)*(m,n) = 0 <==> 16m + 9n = 0. This equation gives m, and

n for which the directional derivative of f at (3,4) in the direction of u = (m,n) vanishes.

As an example, we can take u = (9,-16).