# What is the derivative of y=1/log_2(x)?

Aug 6, 2015

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

#### Explanation:

Note that:

$\frac{1}{\log} _ 2 x = \log \frac{2}{\log} \left(x\right)$

We then have using the chain rule:

$y = k f \left(g \left(x\right)\right)$
$y ' = k f ' \left(g \left(x\right)\right) g ' \left(x\right)$
$f \left(u\right) = \frac{1}{u}$
$g \left(x\right) = \log x$

These have derivatives:
$f ' \left(u\right) = - \frac{1}{u} ^ 2$
$g ' \left(x\right) = \frac{1}{x}$

The full derivative is therefore:
$y ' = - \log \frac{2}{x {\left(\log x\right)}^{2}}$