User:Gauge00/1/f

$1/f(x)$ is the multiplicative inverse of f(x).

1/x

$1/x$ is the multiplicative inverse of $f(x)=x$ .

It is the derivative of the $ln(x)$ , the logarithm function.

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

Its derivatives are

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

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

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

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

xy=1

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

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

we got a hyperbola equation.

$1/f(x)$ is the multiplicative inverse of $f(x)$ .