Cross partial with implicit differentiation

Question

For some $x$ defined by an implicit function, i'm interested in how $\frac{\partial x}{\partial k}$ changes with $x$. Is this simply

$$\frac{\partial^2 x}{\partial k \partial x}$$

Or does this not make sense?

Answer 1

Let $x:=k^2+p^2$. Then: $$\frac{\partial x}{\partial k}=2k$$ Now derivate it with respect to $x$: $$\frac{\partial(2k)}{\partial x}=\frac{\partial(2k)}{\partial k}\frac{\partial k}{\partial x}=2\frac{\partial k}{\partial x}$$ So I think it makes sense.