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)?
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) * limt->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?
Also/anyway, f(r) and r(x,y) are differentiable so you can apply the chain rule...
... where (key in spoiler) ...
Similarly with dh/dy. Then substitute into the left-hand side of the show-sentence.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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