The gradient is $\displaystyle \bigtriangledown f (x,y)= (2y, 2x)$, and at $\displaystyle (1,1)$ we have $\displaystyle \bigtriangledown f (1,1)= (2, 2)$. The directional derivative in the direction of the unit vector $\displaystyle u=(a,b)$ is $\displaystyle \bigtriangledown f (1,1) \cdot (a,b)=2a+2b$. So if you know that this directional derivative is equal to some value $\displaystyle X$, you want to solve the system of equations:

$\displaystyle 2a+2b=X$

$\displaystyle a^2+b^2=1$

for $\displaystyle a,b$.