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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 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 ...
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiable multivariable function f : R n → R n is invertible in a neighborhood of a point p as long as the Jacobian matrix of f at p is invertible .
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).
The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure). For example, consider : ...
The slope field of () = +, showing three of the infinitely many solutions that can be produced by varying the arbitrary constant c.. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral [Note 1] of a continuous function f is a differentiable function F whose derivative is equal to the original function f.
The above theorem generalizes in the obvious way to holomorphic functions: Let and be two open and simply connected sets of , and assume that : is a biholomorphism. Then f {\displaystyle f} and f − 1 {\displaystyle f^{-1}} have antiderivatives, and if F {\displaystyle F} is an antiderivative of f {\displaystyle f} , the general antiderivative ...
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 ...