Derivatives and differentiability.

Let f(x,y)= (xy² - x³)/(x²+y²) where (x,y)!= (0,0) and 0 where (x, y) = (0, 0).

a. Find f_x(a, b) and f_y(a, b) for (a, b) != (0, 0) (Here (a,b) is not equal to (0,0).)

b. Hence explain why f is differentiable at (a, b) != (0, 0). (Here (a,b) is not equal to (0,0).)

c. Find D_uf(a,b) where (a,b) != (0,0) and u = (u₁,u₂) is a unit vector.

d. Find the direction (as a unit vector) at which f increases most rapidly at the point (1, 0). What is this greatest rate of increase?

e. Find, using the definition of the directional derivative as a limit, D_uf(0,0) where u = (u₁,u₂) is a unit vector.

f. Hence find f(0,0).

g. Is f differentiable at (0, 0)? Explain.

h. Is f continuous at (0, 0)? Explain.