Derivatives of Inverse Functions

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

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

In mathematics, the inverse of a function $y=f(x)$ is a function that, in some fashion, "undoes" the effect of $f$. 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:

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

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

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

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):

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

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

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

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

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

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

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

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

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.

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx} = e^x \mbox{ }\mbox{ }\mbox{ }\mbox{ }; \mbox{ }\mbox{ }\mbox{ }\mbox{ } \frac{dx}{dy} = \frac{1}{y} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{dx}\,\cdot\,\frac{dx}{dy} = e^x \cdot \frac{1}{y} = \frac{e^x}{e^x} = 1 }

Resources

• Derivatives of Inverse Functions PowerPoint file created by Dr. Sara Shirinkam, UTSA.

• Derivatives of Inverse Functions Worksheet

• Derivatives of Inverse Function, Mathematics LibreTexts

Licensing

Content obtained and/or adapted from:

• Inverse functions and differentiation, Wikipedia under a CC BY-SA license