Inverse functions and differentiation
Encyclopedia
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 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 reflection
s, 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.
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.
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
so that
,
which agrees with the direct calculation.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the inverse of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
is a function that, in some fashion, "undoes" the effect of (see inverse functionInverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
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
Leibniz notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" increments of x and y, just as Δx and Δy represent finite increments of x and y...
suggests; that is:
This is a direct consequence of the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
, since
and the derivative of
with respect to 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 reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s, in the line y=x. This reflection operation turns the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of any line into its reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
.
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.
Examples
-
(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.
-
has inverse 
(for positive 
)
Additional properties
- 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 functions with continuousContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
derivative have inverses in a neighbourhoodNeighbourhood (mathematics)In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of every point where the derivative is non-zero. This need not be true if the derivative is not continuous.
Higher derivatives
The chain ruleChain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
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
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives, named after , though he was not the first to state or prove the formula...
.
These formulas can also be written using Lagrange's notation. If f and g are inverses, then
Example
-
has the inverse 
. Using the formula for the second derivative of the inverse function,
so that
,which agrees with the direct calculation.
See also
- CalculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
- Inverse functionInverse functionIn mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
s - Chain ruleChain ruleIn calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
- Inverse function theoremInverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
- Implicit function theoremImplicit function theoremIn multivariable calculus, the implicit function theorem is a tool which allows relations to be converted to functions. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on...