In mathematics, the inverse of a function
is a function that, in some fashion, "undoes" the effect of
. The inverse of
is denoted as
, where
if and only if
.
Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests; that is:
![{\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f95e8b8db53babeadfae565759a5d9b5607efea8)
This relation is obtained by differentiating the equation
in terms of
and applying the chain rule, yielding that:
![{\displaystyle {\frac {dx}{dy}}\,\cdot \,{\frac {dy}{dx}}={\frac {dx}{dx}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4121e1bf0f01fa996b2bd4e831762cbed4ea40)
considering that the derivative of
with respect to
is 1.
Writing explicitly the dependence of
on
, and the point at which the differentiation takes place, the formula for the derivative of the inverse becomes (in Lagrange's notation):
.
This formula holds in general whenever
is continuous and injective on an interval
, with
being differentiable at
(
) and where
. 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)}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b71cf74aa5480ff9d43a9823ceb5580914f923a5)
where
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
. This reflection operation turns the gradient of any line into its reciprocal.
Assuming that
has an inverse in a neighbourhood of
and that its derivative at that point is non-zero, its inverse is guaranteed to be differentiable at
and have a derivative given by the above formula.
Examples
(for positive x) has inverse
.
![{\displaystyle {\frac {dy}{dx}}=2x{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{2{\sqrt {y}}}}={\frac {1}{2x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52485cf58c23be2bcea9246cfedd3b6c2775d142)
![{\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=2x\cdot {\frac {1}{2x}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a37029e7ca0d81e754a7d58d14d812100a698d)
At
, however, there is a problem: the graph of the square root function becomes vertical, corresponding to a horizontal tangent for the square function.
(for real x) has inverse
(for positive
)
![{\displaystyle {\frac {dy}{dx}}=e^{x}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }};{\mbox{ }}{\mbox{ }}{\mbox{ }}{\mbox{ }}{\frac {dx}{dy}}={\frac {1}{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a637f8a50643b148f8cece06e79a882aa254c3)
![{\displaystyle {\frac {dy}{dx}}\,\cdot \,{\frac {dx}{dy}}=e^{x}\cdot {\frac {1}{y}}={\frac {e^{x}}{e^{x}}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1754e0ae192bedb64a5ce9dc957feaac9cc2910f)
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