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

Dec 26, 2015

Apply the product rule to find:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \left(\cos x + 1\right) \left(x + {e}^{x}\right) + \left(\sin x + x\right) \left(1 + {e}^{x}\right)$

#### Explanation:

The product rule tells us that:

$\frac{d}{\mathrm{dx}} u \left(x\right) v \left(x\right) = u ' \left(x\right) v \left(x\right) + u \left(x\right) v ' \left(x\right)$

So with $u \left(x\right) = \left(\sin x + x\right)$ and $v \left(x\right) = \left(x + {e}^{x}\right)$, we find:

$\frac{d}{\mathrm{dx}} f \left(x\right) = \left(\cos x + 1\right) \left(x + {e}^{x}\right) + \left(\sin x + x\right) \left(1 + {e}^{x}\right)$