# What is the derivative of f(x)=log(x)/x ?

Aug 14, 2014

The derivative is $f ' \left(x\right) = \frac{1 - \log x}{x} ^ 2$.

This is an example of the the Quotient Rule:

The quotient rule states that the derivative of a function $f \left(x\right) = \frac{u \left(x\right)}{v \left(x\right)}$ is:

$f ' \left(x\right) = \frac{v \left(x\right) u ' \left(x\right) - u \left(x\right) v ' \left(x\right)}{v \left(x\right)} ^ 2$.

To put it more concisely:

$f ' \left(x\right) = \frac{v u ' - u v '}{v} ^ 2$, where $u$ and $v$ are functions (specifically, the numerator and denominator of the original function $f \left(x\right)$).

For this specific example, we would let $u = \log x$ and $v = x$. Therefore $u ' = \frac{1}{x}$ and $v ' = 1$.

Substituting these results into the quotient rule, we find:

$f ' \left(x\right) = \frac{x \times \frac{1}{x} - \log x \times 1}{x} ^ 2$

$f ' \left(x\right) = \frac{1 - \log x}{x} ^ 2$.