Inverse functions and differentiation

Short description: Calculus identity

Rule:
f′(x)=1(f−1)′(f(x))

Example for arbitrary x0≈5.8:
f′(x0)=14
(f−1)′(f(x0))=4 Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

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* t
* e

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 as f−1, where f−1(y)=x if and only if f(x)=y.

Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:

dxdy⋅dydx=1.

This relation is obtained by differentiating the equation f−1(y)=x in terms of x and applying the chain rule, yielding that:

dxdy⋅dydx=dxdx

considering that 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, the formula for the derivative of the inverse becomes (in Lagrange's notation):

[f−1](a)=1f′(f−1(a)).

This formula holds in general whenever f is continuous and injective on an interval I, with f being differentiable at f−1(a)(∈I) and wheref′(f−1(a))≠0. The same formula is also equivalent to the expression

𝒟[f−1]=1(𝒟f)∘(f−1),

where 𝒟 denotes the unary derivative operator (on the space of functions) and ∘ denotes function composition.

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.[1]

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

* 1 Examples
* 2 Additional properties
* 3 Higher derivatives
* 4 Example
* 5 See also
* 6 References

## Examples

* y=x2 (for positive x) has inverse x=y.

dydx=2x ; dxdy=12y=12x

dydx⋅dxdy=2x⋅12x=1.

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=ex (for real x) has inverse x=ln⁡y (for positive y)

dydx=ex ; dxdy=1y

dydx⋅dxdy=ex⋅1y=exex=1

## Additional properties

* Integrating this relationship gives

f−1(x)=∫1f′(f−1(x))dx+C.

This is only useful if the integral exists. In particular we need f′(x) 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.

* Another very interesting and useful property is the following:

∫f−1(x)dx=xf−1(x)−F(f−1(x))+C

Where F denotes the antiderivative of f.

## Higher derivatives

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

d2ydx2⋅dxdy+ddx(dxdy)⋅(dydx)=0,

that is simplified further by the chain rule as

d2ydx2⋅dxdy+d2xdy2⋅(dydx)2=0.

Replacing the first derivative, using the identity obtained earlier, we get

d2ydx2=−d2xdy2⋅(dydx)3.

Similarly for the third derivative:

d3ydx3=−d3xdy3⋅(dydx)4−3d2xdy2⋅d2ydx2⋅(dydx)2

or using the formula for the second derivative,

d3ydx3=−d3xdy3⋅(dydx)4+3(d2xdy2)2⋅(dydx)5

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

g″(x)=−f″(g(x))[f′(g(x))]3

## Example

* y=ex has the inverse x=ln⁡y. Using the formula for the second derivative of the inverse function,

dydx=d2ydx2=ex=y ; (dydx)3=y3;

so that

d2xdy2⋅y3+y=0 ; d2xdy2=−1y2,

which agrees with the direct calculation.

## See also

* Calculus
* Inverse functions
* Chain rule
* Inverse function theorem
* Implicit function theorem
* Integration of inverse functions

## References

1. ↑ "Derivatives of Inverse Functions". https://oregonstate.edu/instruct/mth251/cq/Stage6/Lesson/inverseDeriv.html.

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Calculus

Precalculus|

* Binomial theorem
* Concave function
* Continuous function
* Factorial
* Finite difference
* Free variables and bound variables
* Fundamental theorem
* Graph of a function
* Linear function
* Mean value theorem
* Radian
* Rolle's theorem
* Secant
* Slope
* Tangent

Limits|

* Indeterminate form
* Limit of a function
* One-sided limit
* Limit of a sequence
* Orders of approximation
* (ε, δ)-definition of limit

Differential calculus|

* Chain rule
* Derivative
* Differential
* Differential equation
* Differential operator
* Implicit differentiation
* Inverse functions and differentiation
* L'Hôpital's rule
* Leibniz's rule
* Logarithmic derivative
* Mean value theorem
* Newton's method
* Notation
* Leibniz's notation
* Newton's notation
* Product rule
* Quotient rule
* Regiomontanus' angle maximization problem
* Related rates
* Simplest rules
* Constant factor rule in differentiation
* Linearity of differentiation
* Power rule
* Sum rule in differentiation
* Stationary point
* First derivative test
* Second derivative test
* Extreme value theorem
* Maxima and minima
* Taylor's theorem

Integral calculus|

* Antiderivative
* Arc length
* Constant of integration
* Differentiation under the integral sign
* Fundamental theorem of calculus
* Integral of secant cubed
* Integral of the secant function
* Integration by parts
* Integration by substitution
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Proof that 22/7 exceeds π
* Simplest rules
* Constant factor rule in integration
* Linearity of integration
* Sum rule in integration
* Trapezoidal rule
* Trigonometric substitution

Vector calculus|

* Curl
* Directional derivative
* Divergence
* Divergence theorem
* Gradient
* Gradient theorem
* Green's theorem
* Laplacian
* Stokes' theorem

Multivariable calculus|

* Curvature
* Disc integration
* Divergence theorem
* Exterior
* Gabriel's Horn
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Shell integration
* Surface integral
* Tensor
* Volume integral

Series|

* Abel's test
* Alternating
* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
* Geometric
* Harmonic
* Infinite
* Integral test for convergence
* Limit comparison test
* Maclaurin
* Power
* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

* Differentiation rules
* List of integrals of exponential functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of inverse trigonometric functions
* List of integrals of irrational functions
* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
* Lists of integrals

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