Inverse functions and differentiation

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In mathematics, the inverse of a function y = f(x) is a function that, in some fashion, "undoes" the effect of f (see inverse function for a formal and detailed definition). The inverse of f is denoted f − 1. 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 x with respect to x is 1.

Writing explicitly the dependence of y on x 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 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 differentiable at x and have a derivative given by the above formula.

Contents

Examples

  • \,y = x^2 (for positive x) has inverse x = \sqrt{y}.

At x=0, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.

  • \,y = e^x has inverse  x = \ln\,y (for positive y)

Additional properties

  • Integrating this relationship gives

Higher derivatives

The chain rule given above is obtained by differentiating the identity x=f -1(f(x)) with respect to x. One can continue the same process for higher derivatives. Differentiating the identity with respect to x two times, one obtains

or replacing the first derivative using the formula above,

Similarly for the third derivative:

or using the formula for the second derivative,

These formulas are generalized by the Faà di Bruno's formula.

These formulas can also be written using Lagrange's notation. If f and g are inverses, then

Example

  • \,y = e^x has the inverse x = \ln\,{y}. Using the formula for the second derivative of the inverse function,

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