# How do you differentiate f(x)= (1 - sin^2x)/(x-cosx)  using the quotient rule?

Jun 11, 2018

$f ' \left(x\right) = \frac{\cos \left(x\right) \left(\cos \left(x\right) \left(\sin \left(x\right) - 1\right) - 2 x \sin \left(x\right)\right)}{x - \cos \left(x\right)} ^ 2$

#### Explanation:

We have
$f ' \left(x\right) = \frac{2 \cos \left(x\right) \left(- \sin \left(x\right)\right) \left(x - \cos \left(x\right) - {\cos}^{2} \left(x\right) \left(1 + \sin \left(x\right)\right)\right)}{x - \cos \left(x\right)} ^ 2$

I have used that

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

since

$1 - {\sin}^{2} \left(x\right) = {\cos}^{2} \left(x\right)$