# How do you differentiate f(x)=(1-lnx)/(1+lnx) using the quotient rule?

Oct 30, 2017

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

#### Explanation:

Using the quotient rule, we have:

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

$\text{ } = \frac{\left(1 + \ln x\right) \left(- \frac{1}{x}\right) - \left(1 - \ln x\right) \left(\frac{1}{x}\right)}{1 + \ln x} ^ 2$

$\text{ } = \frac{\left(- \frac{1}{x}\right) \left(1 + \ln x + 1 - \ln x\right)}{1 + \ln x} ^ 2$

$\text{ } = \frac{\left(- \frac{1}{x}\right) \left(2\right)}{1 + \ln x} ^ 2$

$\text{ } = - \frac{2}{x {\left(1 + \ln x\right)}^{2}}$