The function is #f(x,y)=(2xy)/(x^2+y^2)#
and #(x,y)!=(0,0)#
The gradient is
#gradf(x,y)=((delf)/(delx),(delf)/(dely))#
#(delf)/(delx)=(2y(x^2+y^2)-2xy(2x))/(x^2+y^2)^2=(2y(-x^2+y^2))/(x^2+y^2)^2#
#(delf)/(dely)=(2x(x^2+y^2)-2xy(2y))/(x^2+y^2)^2=(2x(x^2-y^2))/(x^2+y^2)^2#
At the point #(0,0)#
#gradf(0,0)# is undefined
#(1)# The direction is #vecu=(0,1)#, #theta=pi/2#
The magnitude of #||vecu||=sqrt(0^2+1^2)=1#
#vecu# is already normalized
#hatu=<0,1>#
The directional derivative is the dot product of the gradient and the unit vector in the specified direction.
# D( (2xy) / (x^2+y^2))_(hatu) (0,0) = < * *, >. < 0,1 > # is undefined
#(2)# The direction is #vecv=(1,1)#, #theta=pi/4#
The magnitude of #||vecv||=sqrt(1^2+1^2)=sqrt2#
The unit vector is #hatv=1/||vecv||vecv=1/sqrt2<1,1>#
