Multivariable Chain Rule Stewart Early Transcendentals 47

Question

Does anyone know how could I approach this? if $z = \frac{1}{x} [f(x - y) + g(x + y)]$, show that $$ \frac{∂}{∂x}(x^2 \frac{∂z}{∂x} )=x^2 \frac{∂^2 z}{∂y^2}$$

I've tried by substituting $u=x-y$ and $v=x+y$

Answer 1

Don't do that.

Rather, let $h(x,y)=f(x-y)+g(x+y)$ so you can work with its partial derivatives and the quotient rule.

$$\begin{align}\dfrac{\partial z}{\partial x}&=\dfrac{\partial ~~}{\partial x}\dfrac{h(x,y)}{x}\\[1ex]&=\dfrac{x~h_x(x,y)-h(x,y)}{x^2}\end{align}$$

And so forth.

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Where $h_{x}(x,y) = f'(x-y)+g'(x+y)$, and $h_y(x,y)=-f'(x-y)+g'(x+y)$, by the chain rule, et cetera.