StellarAn inverse function reverses the effect of an original function. If f and f-1 are inverse functions, then applying one after the other returns the original input. The derivative relationship for inverse functions is whenever the derivative exists and the denominator is not zero. This means the derivative of an inverse function can be found using the derivative of the original function. Another way to understand this is through the slope relationship: if a function and its inverse swap x and y, then their tangent slopes are reciprocals of each other at corresponding points.
You can also differentiate inverse relationships by using implicit differentiation. If , then applying f to both sides gives f(y)=x, and differentiating that equation often leads directly to the inverse derivative formula.