Inverse function rule

In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of ${\displaystyle f}${\displaystyle f} is denoted as ${\displaystyle f^{-1}}${\displaystyle f^{-1}}, where ${\displaystyle f^{-1}(y)=x}${\displaystyle f^{-1}(y)=x} if and only if ${\displaystyle f(x)=y}${\displaystyle f(x)=y}, then the inverse function rule is, in Lagrange's notation,

${\displaystyle \left[f^{-1}\right]'(y)={\frac {1}{f'\left(f^{-1}(y)\right)}}}${\displaystyle \left[f^{-1}\right]'(y)={\frac {1}{f'\left(f^{-1}(y)\right)}}}.

This formula holds in general whenever ${\displaystyle f}${\displaystyle f} is continuous and injective on an interval I, with ${\displaystyle f}${\displaystyle f} being differentiable at ${\displaystyle f^{-1}(y)}${\displaystyle f^{-1}(y)}(${\displaystyle \in I}${\displaystyle \in I}) and where${\displaystyle f'(f^{-1}(y))\neq 0}${\displaystyle f'(f^{-1}(y))\neq 0}. The same formula is also equivalent to the expression

${\displaystyle {\mathcal {D}}\left[f^{-1}\right]={\frac {1}{({\mathcal {D}}f)\circ \left(f^{-1}\right)}},}${\displaystyle {\mathcal {D}}\left[f^{-1}\right]={\frac {1}{({\mathcal {D}}f)\circ \left(f^{-1}\right)}},}

where ${\displaystyle {\mathcal {D}}}${\displaystyle {\mathcal {D}}} denotes the unary derivative operator (on the space of functions) and ${\displaystyle \circ }${\displaystyle \circ } denotes function composition.

Geometrically, a function and inverse function have graphs that are reflections, in the line ${\displaystyle y=x}${\displaystyle y=x}. This reflection operation turns the gradient of any line into its reciprocal.^[1]

Assuming that ${\displaystyle f}${\displaystyle f} has an inverse in a neighbourhood of ${\displaystyle x}${\displaystyle x} and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at ${\displaystyle x}${\displaystyle x} and have a derivative given by the above formula.

The inverse function rule may also be expressed in Leibniz's notation. As that notation suggests,

${\displaystyle {\frac {dx}{dy}},円\cdot ,円{\frac {dy}{dx}}=1.}${\displaystyle {\frac {dx}{dy}},円\cdot ,円{\frac {dy}{dx}}=1.}

This relation is obtained by differentiating the equation ${\displaystyle f^{-1}(y)=x}${\displaystyle f^{-1}(y)=x} in terms of x and applying the chain rule, yielding that:

${\displaystyle {\frac {dx}{dy}},円\cdot ,円{\frac {dy}{dx}}={\frac {dx}{dx}}}${\displaystyle {\frac {dx}{dy}},円\cdot ,円{\frac {dy}{dx}}={\frac {dx}{dx}}}

considering that the derivative of x with respect to x is 1.

Let ${\displaystyle f}${\displaystyle f} be an invertible (bijective) function, let ${\displaystyle x}${\displaystyle x} be in the domain of ${\displaystyle f}${\displaystyle f}, and let ${\displaystyle y=f(x).}${\displaystyle y=f(x).} Let ${\displaystyle g=f^{-1}.}${\displaystyle g=f^{-1}.} So, ${\displaystyle f(g(y))=y.}${\displaystyle f(g(y))=y.} Derivating this equation with respect to ⁠${\displaystyle y}${\displaystyle y}⁠, and using the chain rule, one gets

${\displaystyle f'(g(y))\cdot g'(y)=1.}${\displaystyle f'(g(y))\cdot g'(y)=1.}

That is,

${\displaystyle g'(y)={\frac {1}{f'(g(y))}}}${\displaystyle g'(y)={\frac {1}{f'(g(y))}}}

or

${\displaystyle (f^{-1})^{\prime }(y)={\frac {1}{f^{\prime }(f^{-1}(y))}}.}${\displaystyle (f^{-1})^{\prime }(y)={\frac {1}{f^{\prime }(f^{-1}(y))}}.}

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

${\displaystyle {\frac {dy}{dx}}=2x{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{2{\sqrt {y}}}}={\frac {1}{2x}}}${\displaystyle {\frac {dy}{dx}}=2x{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{2{\sqrt {y}}}}={\frac {1}{2x}}}

${\displaystyle {\frac {dy}{dx}},円\cdot ,円{\frac {dx}{dy}}=2x\cdot {\frac {1}{2x}}=1.}${\displaystyle {\frac {dy}{dx}},円\cdot ,円{\frac {dx}{dy}}=2x\cdot {\frac {1}{2x}}=1.}

At ${\displaystyle x=0}${\displaystyle 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.

• ${\displaystyle y=e^{x}}${\displaystyle y=e^{x}} (for real x) has inverse ${\displaystyle x=\ln {y}}${\displaystyle x=\ln {y}} (for positive ${\displaystyle y}${\displaystyle y})

${\displaystyle {\frac {dy}{dx}}=e^{x}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{y}}=e^{-x}}${\displaystyle {\frac {dy}{dx}}=e^{x}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{y}}=e^{-x}}

${\displaystyle {\frac {dy}{dx}},円\cdot ,円{\frac {dx}{dy}}=e^{x}\cdot e^{-x}=1.}${\displaystyle {\frac {dy}{dx}},円\cdot ,円{\frac {dx}{dy}}=e^{x}\cdot e^{-x}=1.}

• Integrating this relationship gives

${\displaystyle {f^{-1}}(x)=\int {\frac {1}{f'({f^{-1}}(x))}},円{dx}+C.}${\displaystyle {f^{-1}}(x)=\int {\frac {1}{f'({f^{-1}}(x))}},円{dx}+C.}

This is only useful if the integral exists. In particular we need ${\displaystyle f'(x)}${\displaystyle 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:

${\displaystyle \int f^{-1}(x),円{dx}=xf^{-1}(x)-F(f^{-1}(x))+C}${\displaystyle \int f^{-1}(x),円{dx}=xf^{-1}(x)-F(f^{-1}(x))+C}

Where ${\displaystyle F}${\displaystyle F} denotes the antiderivative of ${\displaystyle f}${\displaystyle f}.

• The inverse of the derivative of f(x) is also of interest, as it is used in showing the convexity of the Legendre transform.

Let ${\displaystyle z=f'(x)}${\displaystyle z=f'(x)} then we have, assuming ${\displaystyle f''(x)\neq 0}${\displaystyle f''(x)\neq 0}:$${\displaystyle {\frac {d(f')^{-1}(z)}{dz}}={\frac {1}{f''(x)}}}$${\displaystyle {\frac {d(f')^{-1}(z)}{dz}}={\frac {1}{f''(x)}}}This can be shown using the previous notation ${\displaystyle y=f(x)}${\displaystyle y=f(x)}. Then we have:

$${\displaystyle f'(x)={\frac {dy}{dx}}={\frac {dy}{dz}}{\frac {dz}{dx}}={\frac {dy}{dz}}f''(x)\Rightarrow {\frac {dy}{dz}}={\frac {f'(x)}{f''(x)}}}$${\displaystyle f'(x)={\frac {dy}{dx}}={\frac {dy}{dz}}{\frac {dz}{dx}}={\frac {dy}{dz}}f''(x)\Rightarrow {\frac {dy}{dz}}={\frac {f'(x)}{f''(x)}}}Therefore:

${\displaystyle {\frac {d(f')^{-1}(z)}{dz}}={\frac {dx}{dz}}={\frac {dy}{dz}}{\frac {dx}{dy}}={\frac {f'(x)}{f''(x)}}{\frac {1}{f'(x)}}={\frac {1}{f''(x)}}}${\displaystyle {\frac {d(f')^{-1}(z)}{dz}}={\frac {dx}{dz}}={\frac {dy}{dz}}{\frac {dx}{dy}}={\frac {f'(x)}{f''(x)}}{\frac {1}{f'(x)}}={\frac {1}{f''(x)}}}

By induction, we can generalize this result for any integer ${\displaystyle n\geq 1}${\displaystyle n\geq 1}, with ${\displaystyle z=f^{(n)}(x)}${\displaystyle z=f^{(n)}(x)}, the nth derivative of f(x), and ${\displaystyle y=f^{(n-1)}(x)}${\displaystyle y=f^{(n-1)}(x)}, assuming ${\displaystyle f^{(i)}(x)\neq 0{\text{ for }}0<i\leq n+1}${\displaystyle f^{(i)}(x)\neq 0{\text{ for }}0<i\leq n+1}:

${\displaystyle {\frac {d(f^{(n)})^{-1}(z)}{dz}}={\frac {1}{f^{(n+1)}(x)}}}${\displaystyle {\frac {d(f^{(n)})^{-1}(z)}{dz}}={\frac {1}{f^{(n+1)}(x)}}}

The chain rule given above is obtained by differentiating the identity ${\displaystyle f^{-1}(f(x))=x}${\displaystyle 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

${\displaystyle {\frac {d^{2}y}{dx^{2}}},円\cdot ,円{\frac {dx}{dy}}+{\frac {d}{dx}}\left({\frac {dx}{dy}}\right),円\cdot ,円\left({\frac {dy}{dx}}\right)=0,}${\displaystyle {\frac {d^{2}y}{dx^{2}}},円\cdot ,円{\frac {dx}{dy}}+{\frac {d}{dx}}\left({\frac {dx}{dy}}\right),円\cdot ,円\left({\frac {dy}{dx}}\right)=0,}

that is simplified further by the chain rule as

${\displaystyle {\frac {d^{2}y}{dx^{2}}},円\cdot ,円{\frac {dx}{dy}}+{\frac {d^{2}x}{dy^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{2}=0.}${\displaystyle {\frac {d^{2}y}{dx^{2}}},円\cdot ,円{\frac {dx}{dy}}+{\frac {d^{2}x}{dy^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{2}=0.}

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

${\displaystyle {\frac {d^{2}y}{dx^{2}}}=-{\frac {d^{2}x}{dy^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{3}.}${\displaystyle {\frac {d^{2}y}{dx^{2}}}=-{\frac {d^{2}x}{dy^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{3}.}

Similarly for the third derivative:

${\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{4}-3{\frac {d^{2}x}{dy^{2}}},円\cdot ,円{\frac {d^{2}y}{dx^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{2}}${\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{4}-3{\frac {d^{2}x}{dy^{2}}},円\cdot ,円{\frac {d^{2}y}{dx^{2}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{2}}

or using the formula for the second derivative,

${\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{4}+3\left({\frac {d^{2}x}{dy^{2}}}\right)^{2},円\cdot ,円\left({\frac {dy}{dx}}\right)^{5}}${\displaystyle {\frac {d^{3}y}{dx^{3}}}=-{\frac {d^{3}x}{dy^{3}}},円\cdot ,円\left({\frac {dy}{dx}}\right)^{4}+3\left({\frac {d^{2}x}{dy^{2}}}\right)^{2},円\cdot ,円\left({\frac {dy}{dx}}\right)^{5}}

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

${\displaystyle g''(x)={\frac {-f''(g(x))}{[f'(g(x))]^{3}}}}${\displaystyle g''(x)={\frac {-f''(g(x))}{[f'(g(x))]^{3}}}}

Higher derivatives of an inverse function can also be expressed with Faà di Bruno's formula and can be written succinctly as:

${\displaystyle \left[f^{-1}\right]^{(n)}(x)=\left[\left({\frac {1}{f'(t)}}{\frac {d}{dt}}\right)^{n}t\right]_{t=f^{-1}(x)}}${\displaystyle \left[f^{-1}\right]^{(n)}(x)=\left[\left({\frac {1}{f'(t)}}{\frac {d}{dt}}\right)^{n}t\right]_{t=f^{-1}(x)}}

From this expression, one can also derive the nth-integration of inverse function with base-point a using Cauchy formula for repeated integration whenever ${\displaystyle f(f^{-1}(x))=x}${\displaystyle f(f^{-1}(x))=x}:

${\displaystyle \left[f^{-1}\right]^{(-n)}(x)={\frac {1}{n!}}\left(f^{-1}(a)(x-a)^{n}+\int _{f^{-1}(a)}^{f^{-1}(x)}\left(x-f(u)\right)^{n},円du\right)}${\displaystyle \left[f^{-1}\right]^{(-n)}(x)={\frac {1}{n!}}\left(f^{-1}(a)(x-a)^{n}+\int _{f^{-1}(a)}^{f^{-1}(x)}\left(x-f(u)\right)^{n},円du\right)}

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

${\displaystyle {\frac {dy}{dx}}={\frac {d^{2}y}{dx^{2}}}=e^{x}=y{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}\left({\frac {dy}{dx}}\right)^{3}=y^{3};}${\displaystyle {\frac {dy}{dx}}={\frac {d^{2}y}{dx^{2}}}=e^{x}=y{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}\left({\frac {dy}{dx}}\right)^{3}=y^{3};}

so that

${\displaystyle {\frac {d^{2}x}{dy^{2}}},円\cdot ,円y^{3}+y=0{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {d^{2}x}{dy^{2}}}=-{\frac {1}{y^{2}}}}${\displaystyle {\frac {d^{2}x}{dy^{2}}},円\cdot ,円y^{3}+y=0{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {d^{2}x}{dy^{2}}}=-{\frac {1}{y^{2}}}},

which agrees with the direct calculation.

• Calculus – Branch of mathematics
• Chain rule – Formula in calculus
• Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function
• Differentiation rules – Rules for computing derivatives of functions
• Implicit function theorem – On converting relations to functions of several real variables
• Integration of inverse functions – Mathematical theorem, used in calculusPages displaying short descriptions of redirect targets
• Inverse function – Mathematical concept
• Inverse function theorem – Theorem in mathematics
• Table of derivatives – Rules for computing derivatives of functionsPages displaying short descriptions of redirect targets
• Vector calculus identities – Mathematical identities

• Marsden, Jerrold E.; Weinstein, Alan (1981). "Chapter 8: Inverse Functions and the Chain Rule". Calculus unlimited (PDF). Menlo Park, Calif.: Benjamin/Cummings Pub. Co. ISBN 0-8053-6932-5.

• Differentiation rules
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