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


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


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


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


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

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.

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