# How do you differentiate f(x)= 5sinx*(1-2x)^2 + cosx*(x+1) using the product rule?

Sep 26, 2017

$\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right) = 5 \cos x {\left(1 - 2 x\right)}^{2} - 20 \sin x \left(1 - 2 x\right) - \sin x \left(x + 1\right) + \cos x$

#### Explanation:

According to product rule
$\frac{d}{\mathrm{dx}} \left(u . v\right) = v \frac{\mathrm{du}}{\mathrm{dx}} + u \frac{\mathrm{dv}}{\mathrm{dx}}$
where u and v are functions of x

$\frac{d}{\mathrm{dx}} \left(5 \sin x . {\left(1 - 2 x\right)}^{2} + \cos x . \left(x + 1\right)\right)$
$= {\left(1 - 2 x\right)}^{2} \frac{d}{\mathrm{dx}} \left(5 \sin x\right) + 5 \sin x \frac{d}{\mathrm{dx}} \left({\left(1 - 2 x\right)}^{2}\right) + \left(x + 1\right) \frac{d}{\mathrm{dx}} \left(\cos x\right) + \cos x \frac{d}{\mathrm{dx}} \left(x + 1\right)$
$= {\left(1 - 2 x\right)}^{2} \left(5 \cos x\right) + 5 \sin x \left(2 \left(1 - 2 x\right) \left(- 2\right)\right) + \left(x + 1\right) \left(- \sin x\right) + \cos x \left(1\right)$
$= 5 \cos x {\left(1 - 2 x\right)}^{2} - 20 \sin x \left(1 - 2 x\right) - \sin x \left(x + 1\right) + \cos x$