Directional derivatives and partial derivatives

Suppose f: R -> R is differentiable and let h(x,y) = f(√(x^2 + y^2)) for x ≠ 0. Letting r = √(x^2 + y^2), show that:

x(dh/dx) + y(dh/dy) = rf'(r).

I have begun by showing that rf'(r) = sqrt(x^2 + y^2) * lim_{t->0} (f(r+t) - f(r))/t

and written out the definition form of the directional derivatives. I cant seem to find a way to equate both sides of the equation. Can anyone help?

Re: Directional derivatives and partial derivatives

Hey tlawrence.

What did you calculate dh/dx and dh/dy in terms of the function f^n(r)? (Note: f^n(r) is the nth derivative of the function f with argument r)?

Re: Directional derivatives and partial derivatives

Also/anyway, f(r) and r(x,y) are differentiable so you can apply the chain rule...

http://www.ballooncalculus.org/draw/diffPartial/six.png

... where (key in spoiler) ...

Similarly with dh/dy. Then substitute into the left-hand side of the show-sentence.

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