User:Gauge00/1/f

${\displaystyle 1/f(x)}$ is the multiplicative inverse of f(x).

${\displaystyle 1/x}$ is the multiplicative inverse of ${\displaystyle f(x)=x}$.

It is the derivative of the ${\displaystyle ln(x)}$, the logarithm function.

${\displaystyle {\frac {1}{x}}={\frac {d}{dx}}\ln(x)}$

Its derivatives are

${\displaystyle {\frac {d}{dx}}\left({\frac {1}{x}}\right)={\frac {-1}{x^{2}}}=-x^{-2}.}$

${\displaystyle {\frac {d^{2}}{d^{2}x}}\left({\frac {1}{x}}\right)={\frac {2}{x^{3}}}=2x^{-3}.}$

${\displaystyle {\frac {d^{3}}{d^{3}x}}\left({\frac {1}{x}}\right)={\frac {-6}{x^{4}}}=-6x^{-4}=-3!x^{-4}.}$

${\displaystyle {\frac {d^{n}}{d^{n}x}}\left({\frac {1}{x}}\right)={\frac {(-1)^{n}n!}{x^{(}n+1)}}=(-1)^{n}n!x^{-(n+1)}}$

The graphes of ${\displaystyle k/x}$ have the form of the hyperbola. For example ${\displaystyle y=1/x}$

${\displaystyle xy=1}$

by changing ${\displaystyle x=x^{\prime }+y^{\prime };y=x^{\prime }-y^{\prime }}$

${\displaystyle {x^{\prime }}^{2}-{y^{\prime }}^{2}=1}$

we got a hyperbola equation.

${\displaystyle 1/f(x)}$ is the multiplicative inverse of ${\displaystyle f(x)}$.

• Inverse (mathematics)