# How do you differentiate y = (sqrt(cos x))/(5lnx)?

May 22, 2018

$- \frac{x \ln \left(x\right) \sin \left(x\right) + 2 \cos \left(x\right)}{10 x \sqrt{\cos \left(x\right)} {\left(\ln \left(x\right)\right)}^{2}}$

#### Explanation:

use quotient rule:

$\frac{d}{\mathrm{dx}} \left(f \frac{x}{g} \left(x\right)\right) = \frac{f ' \left(x\right) g \left(x\right) - g ' \left(x\right) f \left(x\right)}{g \left(x\right)} ^ 2$

(see this proof or this)

using above format for this question, $f \left(x\right) = \sqrt{\cos x}$ and $g \left(x\right) = 5 \ln \left(x\right)$

$y ' \left(x\right) = \frac{\frac{d}{\mathrm{dx}} \left(\sqrt{\cos \left(x\right)}\right) \left(5 \ln \left(x\right)\right) - \frac{d}{\mathrm{dx}} \left(5 \ln \left(x\right)\right) \sqrt{\cos \left(x\right)}}{5 \ln \left(x\right)} ^ 2$

$y ' \left(x\right) = \frac{\frac{1}{2 \sqrt{\cos \left(x\right)}} \cdot \frac{d}{\mathrm{dx}} \left(\cos \left(x\right)\right) \left(5 \ln \left(x\right)\right) - \left(\frac{5}{x}\right) \sqrt{\cos \left(x\right)}}{5 \ln \left(x\right)} ^ 2$

$y ' \left(x\right) = \frac{\frac{1}{2 \sqrt{\cos \left(x\right)}} \cdot \left(- \sin \left(x\right)\right) \left(5 \ln \left(x\right)\right) - \left(\frac{5}{x}\right) \sqrt{\cos \left(x\right)}}{5 \ln \left(x\right)} ^ 2$

when you simplify you should get something like:

$- \frac{x \ln \left(x\right) \sin \left(x\right) + 2 \cos \left(x\right)}{10 x \sqrt{\cos \left(x\right)} {\left(\ln \left(x\right)\right)}^{2}}$