# How do you differentiate f(x)= ( x +2sinx )/ ( x - 6) using the quotient rule?

Jul 14, 2018

$f ' \left(x\right) = - \frac{10 \cos x + 2 \sin x + 6}{x - 6} ^ 2$

#### Explanation:

Here ,

$f \left(x\right) = \frac{x + 2 \sin x}{x - 6}$

Using Quotient Rule ,we get

$f ' \left(x\right) = \frac{\left(x - 6\right) \frac{d}{\mathrm{dx}} \left(x + 2 \sin x\right) - \left(x + 2 \sin x\right) \frac{d}{\mathrm{dx}} \left(x - 6\right)}{x - 6} ^ 2$

$\implies f ' \left(x\right) = \frac{\left(x - 6\right) \left(1 + 2 \cos x\right) - \left(x + 2 \sin x\right) \left(1\right)}{x - 6} ^ 2$

$\implies f ' \left(x\right) = \frac{x + 2 x \cos x - 6 - 12 \cos x - x - 2 \sin x}{x - 6} ^ 2$

$\implies f ' \left(x\right) = \frac{- 10 \cos x - 2 \sin x - 6}{x - 6} ^ 2$

$\implies f ' \left(x\right) = - \frac{10 \cos x + 2 \sin x + 6}{x - 6} ^ 2$