Inverse Functions And Differentiation
In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of (see inverse function for a formal and detailed definition). The inverse of is denoted . The statements y = f(x) and x = f −1(y) are equivalent.
Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:
This is a direct consequence of the chain rule, since
and the derivative of with respect to is 1.
Writing explicitly the dependence of on and the point at which the differentiation takes place and using Lagrange's notation, the formula for the derivative of the inverse becomes
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 has an inverse in a neighbourhood of and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at and have a derivative given by the above formula.
Read more about Inverse Functions And Differentiation: Examples, Additional Properties, Higher Derivatives, Example
Famous quotes containing the words inverse and/or functions:
“Yet time and space are but inverse measures of the force of the soul. The spirit sports with time.”
—Ralph Waldo Emerson (1803–1882)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)