So let's say I have a second order partial derivative of a function with respect to $t$. Now I define $$T=t + f(r)$$ and require $\partial_t^2$ as a function of $\partial_r^2$ and $\partial_T^2$.
Is the second order chain rule for ordinary derivatives applicable here? Or is there any other formula that needs to be applied?
Edit: For some $\phi(t)$, I have attempted the following calculations based on Ted Shifrin's comment $$ \frac{\partial \phi}{\partial t}= \frac{\partial \phi}{\partial T} + f'(r)\frac{\partial \phi}{\partial r} $$ $$ \frac{\partial}{\partial t}\frac{\partial \phi}{\partial t}=\frac{\partial}{\partial t} \left(\frac{\partial \phi}{\partial T}\right) + \frac{\partial}{\partial t} \left[f'(r)\frac{\partial \phi}{\partial r}\right]$$ $$ \frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial T^2} \frac{\partial T}{\partial t} + f''(r)\left(\frac{\partial r}{\partial t}\right) \frac{\partial \phi}{\partial t} + f'(r) \frac{\partial^2 \phi}{\partial r^2} \left(\frac{\partial r}{\partial t }\right)$$