Partial derivatives Q involving homogeneity of degree n

[kostoglotov]

Homework Statement

Show that if f is homogeneous of degree n, then

$$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y)$$

Hint: use the Chain Rule to diff. f(tx,ty) wrt t.

2. The attempt at a solution

I know that if f is homogeneous of degree n then $$t^nf(x,y) = f(tx,ty)$$

But I'm really at a conceptual loss here.

I've tried the hint, I let f(tx, ty) = f(a,b) so a = tx and b = ty then

$$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}\frac{\partial a}{\partial t}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial t} \\<br /> \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}x + \frac{\partial f}{\partial b}y$$

And I have tried looking at

$$\frac{d}{dt}t^nf(x,y) = nt^{n-1}f(x,y)$$

But beyond this I just cannot see what I need to do.

Thanks.

 

[BiGyElLoWhAt]

http://en.wikipedia.org/wiki/Homogeneous_function
Hey! welcome to PF, just so you don't get left hanging for too long, try checking this out.

 

[wabbit]

That's a good start. I'll call the first equation in your attempt (1) and the last one (2).
Now in (2) you are taking the derivative (with respect to t) of the left hand of (1). Do the same thing with the right hand and see where that leads you.