# Question #1f6c9

##### 1 Answer
Aug 18, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{x \ln \left(x\right)}$

#### Explanation:

We have $y \left(u \left(x\right)\right)$ so need to use the chain rule:

$u \left(x\right) = - \frac{1}{\ln} \left(x\right)$

Using the quotient rule:

$\implies \frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{x {\ln}^{2} \left(x\right)}$

$y = \ln \left(u\right) \implies \frac{\mathrm{dy}}{\mathrm{du}} = \frac{1}{u} = - \ln \left(x\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \ln \left(x\right) \cdot \frac{1}{x {\ln}^{2} \left(x\right)} = - \frac{1}{x \ln \left(x\right)}$