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In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 = x {\displaystyle f^{-1}=x} if and only if f = y {\displaystyle f=y} , then the inverse function rule is, in Lagrange's notation,
This formula holds in general whenever f {\displaystyle f} is continuous and injective on an interval I, with f {\displaystyle f} being differentiable at f − 1 {\displaystyle f^{-1}} and where f ′ ] ≠ 0 {\displaystyle f']\neq 0}. The same formula is also equivalent to the expression
where D {\displaystyle {\mathcal {D}}} denotes the unary derivative operator and ∘ {\displaystyle \circ } denotes function composition.
Geometrically, a function and inverse function have graphs that are reflections, in the line y = x {\displaystyle y=x}. This reflection operation turns the gradient of any line into its reciprocal.