# How do you differentiate f(x)=(lnx+x)(1+e^x) using the product rule?

Jan 5, 2016

• Product rule: $\left(a b\right) ' = a ' b + a b '$
$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = \left(\frac{1}{x} + 1\right) \left(1 + {e}^{x}\right) + \left(\ln x + x\right) \left({e}^{x}\right)$
$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = \left({e}^{x} / x + {e}^{x} + \frac{1}{x} + 1\right) + \left({e}^{x} \ln x + x {e}^{x}\right)$
$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = {e}^{x} \left(1 + \frac{1}{x} + \ln x + x\right) + \frac{1}{x} + 1$