Can you calculate $\frac{dx}{dy}$ by finding $\frac{dy}{dx}$ and flipping it over? This may be a silly question but for example if you had the gradient at $x=4$ of $y=x^2+1$, then can you just calculate $\frac{dx}{dy}$ by finding $\frac{dy}{dx}$ and flipping it over? Or must you make $x$ the subject and differentiate?
If $f$ is invertible and differentiable, with $f'$ never zero, then $f^{-1}$ is also differentiable and

$$(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}. $$

Does this help?
Yes, you can. If both derivatives exist, they are reciprocal, that is

$\frac{dy}{dx}\frac{dx}{dy}=1$