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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 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near = (), and the derivative of the inverse function at is the reciprocal of the derivative of at : ′ = ′ = ′ (()).
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective , and if it exists, is denoted by f − 1 . {\displaystyle f^{-1}.}
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations can be ...
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
A function is invertible if and only if its converse relation is a function, in which case the converse relation is the inverse function. The converse relation of a function f : X → Y {\displaystyle f:X\to Y} is the relation f − 1 ⊆ Y × X {\displaystyle f^{-1}\subseteq Y\times X} defined by the graph f − 1 = { ( y , x ) ∈ Y × X : y ...
In particular, the function f has a differentiable inverse function in a neighborhood of a point x if and only if the Jacobian determinant is nonzero at x (see inverse function theorem for an explanation of this and Jacobian conjecture for a related problem of global invertibility).
Nevertheless, they all assume that f or f −1 is differentiable. The general version of the theorem, free from this additional assumption, was proposed by Michael Spivak in 1965, as an exercise in the Calculus, [2] and a fairly complete proof following the same lines was published by Eric Key in 1994. [3]
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