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

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

#### Explanation:

The given function"

f(x)=\frac{\sin x}{e^x+x

Differentiating above function w.r.t. $x$ using quotient rule as follows

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left(\setminus \frac{\setminus \sin x}{{e}^{x} + x}\right)$

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

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

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