Basic partial differentiation help (needs checking)

[niekehecv]

Homework Statement

given z=yf(x^2-y^2)
show that the x(∂z/∂y)+y(∂z/∂x)=xz/y

The Attempt at a Solution

cut it short, my
∂z/∂y= f(x^2-y^2)-2(y^2)f(x^2-y^2)
∂z/∂x=2xyf(x^2-y^2)

i was able to prove that
x(∂z/∂y)+y(∂z/∂x)=xz/y

But i need help with partial differentiations when they give an equation like z=f(x^2-y^2)
I've read about partially differentiating such equations somewhere before. Can someone please check if i am doing it right? Also, what is this kind of partial differentiation called? (such as partially differentiating z=f(x^2-y^2)
I would really appreciate if someone could tell me what is it called so i could read up more about it and do more examples of this kind.

 

[LCKurtz]

When you have something like $z=f(x^2-y^2)$ you need to use the chain rule. The easiest way to see this is to look at as $z = f(u),\ u=x^2-y^2$ Now if you want to calculate $z_x$ you use$$
z_x = f'(u)u_x = f'(x^2-y^2)(-2x)$$You are missing the primes in your argument.

 

[NascentOxygen]

i was able to prove that
x(∂z/∂y)+y(∂z/∂x)=xz/y

I don't think wolframalpha is going to get the answer xz/y. Here's a good start (I think):
http://m.wolframalpha.com/input/?i=(d/dx+y.f(x^2+-+y^2)),+(d/dy+y.f(x^2+-+y^2))&x=0&y=0&js=off

But i need help with partial differentiations when they give an equation like z=f(x^2-y^2)

see above