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.
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
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.
- (for positive ) has inverse .
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.
- (for real ) has inverse (for positive )
- Integrating this relationship gives
- This is only useful if the integral exists. In particular we need to be non-zero across the range of integration.
- It follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.
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. Twice differentiating the identity with respect to x 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
- has the inverse . Using the formula for the second derivative of the inverse function,
which agrees with the direct calculation.