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By the inverse function theorem, a continuous function of a single variable (where ) is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima). For example, the function. is invertible, since the derivative f′(x) = 3x2 + 1 is always positive.
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 : ′ = ′ = ′ (()).
Miscellanea. v. t. e. In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse of a continuous and invertible function , in terms of and an antiderivative of . This formula was published in 1905 by Charles-Ange Laisant. [1]
Calculus. In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can loosely be thought of as using the chain rule "backwards."
In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, [1] cyclometric, [2] or arcus functions [3]) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, [4 ...
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 this: Since the inverse function of f(w) = e w is called the logarithm, it makes sense to call the inverse "function" of the product we w as "product logarithm".
Since the function f(n) = A(n, n) considered above grows very rapidly, its inverse function, f −1, grows very slowly. This inverse Ackermann function f −1 is usually denoted by α. In fact, α(n) is less than 5 for any practical input size n, since A(4, 4) is on the order of .