Inverse Functions and Differentiation
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Inverse Functions and Differentiation
Rule:
${\displaystyle {\color {CornflowerBlue}{f'}}(x)={\frac {1}{{\color {Salmon}{(f^{-1})'}}({\color {Blue}{f}}(x))}}}$

Example for arbitrary ${\displaystyle x_{0}\approx 5.8}$:
${\displaystyle {\color {CornflowerBlue}{f'}}(x_{0})={\frac {1}{4}}}$
${\displaystyle {\color {Salmon}{(f^{-1})'}}({\color {Blue}{f}}(x_{0}))=4~}$

In mathematics, the inverse of a function ${\displaystyle y=f(x)}$ is a function that, in some fashion, "undoes" the effect of ${\displaystyle f}$ (see inverse function for a formal and detailed definition). The inverse of ${\displaystyle f}$ is denoted as ${\displaystyle f^{-1}}$, where ${\displaystyle f^{-1}(y)=x}$ if and only if ${\displaystyle f(x)=y}$.

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

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

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

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

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):

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

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

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

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

Geometrically, a function and inverse function have graphs that are reflections, in the line ${\displaystyle y=x}$. This reflection operation turns the gradient of any line into its reciprocal.[2]

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

## Examples

• ${\displaystyle y=x^{2}}$ (for positive x) has inverse ${\displaystyle x={\sqrt {y}}}$.
${\displaystyle {\frac {dy}{dx}}=2x{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{2{\sqrt {y}}}}={\frac {1}{2x}}}$
${\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=2x\cdot {\frac {1}{2x}}=1.}$

At ${\displaystyle x=0}$, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.

• ${\displaystyle y=e^{x}}$ (for real x) has inverse ${\displaystyle x=\ln {y}}$ (for positive ${\displaystyle y}$)
${\displaystyle {\frac {dy}{dx}}=e^{x}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{y}}}$
${\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=e^{x}\cdot {\frac {1}{y}}={\frac {e^{x}}{e^{x}}}=1}$

${\displaystyle {f^{-1}}(x)=\int {\frac {1}{f'({f^{-1}}(x))}}\,{dx}+C.}$
This is only useful if the integral exists. In particular we need ${\displaystyle f'(x)}$ 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:
${\displaystyle \int f^{-1}(x)\,{dx}=xf^{-1}(x)-F(f^{-1}(x))+C}$
Where ${\displaystyle F}$ denotes the antiderivative of ${\displaystyle f}$.

## Higher derivatives

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

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

that is simplified further by the chain rule as

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

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

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

Similarly for the third derivative:

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

or using the formula for the second derivative,

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

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

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

## Example

• ${\displaystyle y=e^{x}}$ has the inverse ${\displaystyle x=\ln y}$. Using the formula for the second derivative of the inverse function,
${\displaystyle {\frac {dy}{dx}}={\frac {d^{2}y}{dx^{2}}}=e^{x}=y{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}\left({\frac {dy}{dx}}\right)^{3}=y^{3};}$

so that

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

which agrees with the direct calculation.

## References

1. ^ "Derivative of Inverse Functions (a.k.a, How to Create Your Own Table of Derivatives)". Math Vault. 2016-02-28. Retrieved .
2. ^ "Derivatives of Inverse Functions". oregonstate.edu. Retrieved .