# Question #183ce

Jan 14, 2017

Given: $f \left(x\right) = \frac{\sin \left(x\right) - x \cos \left(x\right)}{x \sin \left(x\right) + \cos \left(x\right)}$

$f \left(x\right)$ is of the form $g \frac{x}{h \left(x\right)}$, therefore, The Quotient Rule applies.

#### Explanation:

let $g \left(x\right) = \sin \left(x\right) - x \cos \left(x\right)$

then $g ' \left(x\right) = \cos \left(x\right) - \cos \left(x\right) + x \sin \left(x\right) = x \sin \left(x\right)$

let $h \left(x\right) = x \sin \left(x\right) + \cos \left(x\right)$

then $h ' \left(x\right) = \sin \left(x\right) + x \cos \left(x\right) - \sin \left(x\right) = x \cos \left(x\right)$

and ${h}^{2} \left(x\right) = {x}^{2} {\sin}^{2} \left(x\right) + 2 x \sin \left(x\right) \cos \left(x\right) + {\cos}^{2} \left(x\right)$

The quotient rule is:

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

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

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

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

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