# How do you differentiate f(x)= ( x +2sinx )/ (x + 4 ) using the quotient rule?

Jun 3, 2016

$f ' \left(x\right) = \frac{2 \left(\cos x \left(x + 4\right) - \sin x + 2\right)}{x + 4} ^ 2$

#### Explanation:

The quotient rule states that the derivative of a function expressible as the quotient of two other functions

$f = \left(\frac{g}{h}\right)$

Has a derivative of

${f}^{'} = \frac{{g}^{'} h - g {h}^{'}}{h} ^ 2$

For $f \left(x\right) = \frac{x + 2 \sin x}{x + 4}$:

We see that $g = x + 2 \sin x$ so ${g}^{'} = 1 + 2 \cos x$ and $h = x + 4$ so ${h}^{'} = 1$.

This gives:

$f ' \left(x\right) = \frac{\left(1 + 2 \cos x\right) \left(x + 4\right) - \left(x + 2 \sin x\right) \left(1\right)}{x + 4} ^ 2$

$f ' \left(x\right) = \frac{x + 4 + 2 x \cos x + 8 \cos x - x - 2 \sin x}{x + 4} ^ 2$

$f ' \left(x\right) = \frac{2 \left(\cos x \left(x + 4\right) - \sin x + 2\right)}{x + 4} ^ 2$