# How do you differentiate f(x)=(x+cosx)/(x^2+3) using the quotient rule?

Nov 10, 2016

$\frac{\left(1 - \sin x\right)}{{x}^{2} + 3} - \frac{\left(x + \cos x\right) 2 x}{{x}^{2} + 3} ^ 2$

#### Explanation:

Quotient rule states:
$\frac{f}{g} = \frac{g f ' - f g '}{g} ^ 2$

the derivative of $f \left(x\right) = x + \cos x$ is:
$f ' \left(x\right) = 1 - \sin x$

the derivative of $g \left(x\right) = {x}^{2} + 3$ is:
$g ' \left(x\right) = 2 x$

Now plug in:

$\frac{\left({x}^{2} + 3\right) \left(1 - \sin x\right) - \left(x + \cos x\right) 2 x}{{x}^{2} + 3} ^ 2$

$\frac{\left({x}^{2} + 3\right) \left(1 - \sin x\right)}{{x}^{2} + 3} ^ 2 - \frac{\left(x + \cos x\right) 2 x}{{x}^{2} + 3} ^ 2$

$\frac{\left(1 - \sin x\right)}{{x}^{2} + 3} - \frac{\left(x + \cos x\right) 2 x}{{x}^{2} + 3} ^ 2$