# What is the derivative of ln(x)/sin(2x)?

Jul 10, 2018

$f ' \left(x\right) = \frac{\sin \left(2 x\right) - 2 x \ln \left(x\right) \cdot \cos \left(2 x\right)}{x {\left(\sin \left(2 x\right)\right)}^{2}}$

#### Explanation:

We Need the Quotient rule

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

$\left(\ln \left(x\right)\right) ' = \frac{1}{x}$
$\left(\sin \left(2 x\right)\right) ' = \cos \left(2 x\right) \cdot 2$

so we get

$\frac{\frac{1}{x} \cdot \sin \left(2 x\right) - \ln \left(x\right) \cdot \cos \left(2 x\right) \cdot 2}{\sin \left(2 x\right)} ^ 2$
multiplying numerator and denominator by $x$ we get

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