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?

implicit differentiation operator d/dy
$y = x^2 + 1$
$d/dy(y) = d/dy(x^2 + 1)$
$1 = 2x dx/dy$   (by the chain rule on the RHS)
$dy / dx = 2x$
as expected
