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

Oct 20, 2016

$f ' \left(x\right) = {e}^{x} \cos x + {e}^{x} \sin x$

Explanation:

Use the product rule $\frac{d}{\mathrm{dx}} \left(u v\right) = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

So, with $f \left(x\right) = {e}^{x} \sin x$ we have:

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