Inverse function rule

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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 is denoted as f−1, where f−1(y)=x if and only if f(x)=y, then the inverse function rule is, in Lagrange's notation, [f−1]′(y)=1f′(f−1(y)). This formula holds in general whenever f is continuous and injective on an interval I, with f being differentiable at f−1(y)(∈I) and wheref′(f−1(y))≠0. The same formula is also equivalent to the expression 𝒟[f−1]=1(𝒟f)∘(f−1), where 𝒟 denotes the unary derivative operator (on the space of functions) and ∘ denotes function composition. Geometrically, a function and inverse function have graphs that are reflections, in the line y=x. This reflection operation turns the gradient of any line into its reciprocal. Assuming that f has an inverse in a neighbourhood of x and that its derivative at that point is non-zero, its inverse is guaranteed to be...

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