If $z=f(u,v)$ is a continuous function with continuous first and second order derivatives, $u=xy \text{and} u=\frac{x}{y}$ , use the multivariable chain rule to show that \[x^2\frac{\partial^2z}{\partial x^2}-y^2\f

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richyw · Answer

if $z=f(u,v)$ is a continuous function with continuous first and second order derivatives, $u=xy\text{ and }u=xy$ , use the multivariable chain rule to show that $$x^2\frac{\partial^2z}{\partial x^2}-y^2\frac{\partial^2z}{\partial y^2}=4uv\frac{\partial^2z}{\partial u \partial v}-2v\frac{\partial z}{\partial v}$$

richyw · Answer

oops mest up again ...second order derivatives, $u=xy \text{ and } v=\frac{x}{y}$ , use the multivariable chain rule to show that...

richyw · Answer

so what is this actuall saying. The grammar in the question implies that the first order derivative is $u=xy$ and the second order derivative is $v=\frac{x}{y}$ but this seems like nonsense to me.