In what direction the function $f(x,y)=e^{x-y}+\sin (x+y^2)$ grows fastest at point $(0,0)$?

The fastest growth is in the direction of the gradient 
\[\nabla f=(f_x,f_y)= (e^{x-y}+\cos (x+y^2), -e^{x-y}+2y\cos (x+y^2)).\]
Hence
\[\nabla f(0,0)=(1+1, -1+0)=(2,-1). \]
So the fastest growth is in the direction $(2,-1)$.