Inverse functions and differentiation


In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of . The inverse of is denoted as, where if and only if.
Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:
This relation is obtained by differentiating the equation in terms of and applying the chain rule, yielding that:
considering that the derivative of with respect to is 1.
Writing explicitly the dependence of on, and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes :
This formula holds in general whenever is continuous and injective on an interval, with being differentiable at and where. The same formula is also equivalent to the expression
where denotes the unary derivative operator and denote the binary composition operator.
Geometrically, a function and inverse function have graphs that are reflections, in the line. 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.

Examples

At, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.
The chain rule given above is obtained by differentiating the identity with respect to. One can continue the same process for higher derivatives. Differentiating the identity twice with respect to ', one obtains
that is simplified further by the chain rule as
Replacing the first derivative, using the identity obtained earlier, we get
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 ' and are inverses, then

Example

so that
which agrees with the direct calculation.