I have not studied physics but I was browsing Carroll's relativity book and randomly stumbled upon the following which I would like to understand mathematically. It says$$"ds^{2} = 0 = - \left( 1 - \frac{2GM}{r} \right) dt^{2} + \left( 1 - \frac{2GM}{r} \right)^{-1} dr^{2},$$from which we see that$$\frac{dt}{dr} = \pm \left( 1 - \frac{2GM}{r} \right)^{-1}."$$
I struggle to understand how this was obtained. For simplicity, call $a = \left( 1 - \frac{2GM}{r} \right)$. It is immediate that$$\frac{dt^2}{dr^2} = a^{-2}$$however I don't see why this is the same as $(\frac{dt}{dr})^2$. Unless I completely misunderstand the notation, mathematically wouldn't $\frac{dt^2}{dr^2}$ mean that we differentiate the function $t^2$ with respect to the variable $r^2$? For example, in calculus if we have a function $f(x)$ then $\frac{df^2}{dx^2}$ is clearly not the same as $(f'(x))^2$. What am I fundamentally misundersanding?