Difficult Second Partial Derivative Where Variable of Differentiation is Difficult to Write

Question

I have $k$ as a function of $t$ and $x$, and I know that \begin{align} d\tau &= dt-\left(\frac{1-k}{2}\right) \, dx \\ dr &= \left(\frac{1+k}{2}\right) \, dx \end{align} and I want to find $\dfrac{\partial^2x}{\partial r^2}$. Since I already know $\dfrac{\partial x}{\partial r}=\dfrac{2}{1+k}$, does it work to say $$\frac{\partial^2x}{\partial r^2}=\frac{\partial (\frac{2}{1+k})}{\partial k}\frac{\partial k}{\partial t}\frac{\partial t}{\partial r}+\frac{\partial (\frac{2}{1+k})}{\partial k}\frac{\partial k}{\partial x}\frac{\partial x}{\partial r} \quad ? $$ If not, how can I find it?