Show that x(df/dx) + y(df/dy)= (x^2)(d^2f/dx^2) + 2xy(d^2f/dxdy) + (y^2)(d^2f/dy^2) = 2f(x,y)?

Question

For the function f(x,y)= {[(x^2)(y^3)]/[(x^2) + (y^2)]} ; (x,y) not = (0,0)
& f(x,y) = 0 ; (x,y)=(0,0)

i'm getting stuck with x(df/dx) + y(df/dy), cause i'm getting it equal to 3f i.e. {[5(x^2)(x^2 + y^2)] - [2(x^2)(x^2 + y^2)]} / [(x^2 + y^2)^2] = 3f

Answer 1

You mean f(x,y) = [(x^2)(y^2)]/[(x^2) + (y^2)]
The first derivatives are straightforwardand. It is easier if you take the log of the equation for f first
logf = 2logx + 2logy - log(x^2+y^2)
Then on taking d/dx and d/dy and adding you easily find that
x(df/dx) + y(df/dy) = 2f
To get the second identity take xd/dx and yd/dy of
x(df/dx) + y(df/dy) = 2f
and add the results.