What have you tried for this question?
(Hint: Something is differentiable if it satisfies the Cauchy-Riemann equations).
f(z) = z Im (z^2) = (x+iy) (2xy) = 2(x^2)y + 2x(y^2)i
determine the points in the complex plane such that f is differentiable.
df/dx=2xy+2y(x+iy), so i*df/dx=2ixy+2iy(x+iy)
This tell us f(z) is differentiable when x=iy.
is it right?