The derivative of the inverse function may be found with the following interesting trick: suppose that

and therefore

Let's differentiate both sides of the equality and reorder everything

Let's see this trick in action with some examples

Suppose we know the derivative of the exponential $\exp$ but not that of the logarithm $\ln$

Suppose we know the derivative of the sinus $\sin$ but not that of the arcsinus $\sin^{-1}$

Suppose we know the derivative of the tangent $\tan$ but not that of the arctangent $\tan^{-1}$