Definition
Inverse Rule
Let
be open intervals and let be a bijective differentiable function with for all . Then the inverse function is differentiable and its derivative at is: In Leibniz notation, with
and :
Intuition
Reciprocal slopes
The derivative of
measures how much changes per unit change in . The inverse swaps the roles of and , so its rate of change is the reciprocal. If rises steeply (large ), then crawls horizontally (small ), and vice versa. Geometrically, the graph of
is the reflection of the graph of across the line . Reflecting a tangent segment swaps its horizontal and vertical components, turning a slope of into .