How do you differentiate f(x)= 5sinx*(1-2x)(x+1)-3 using the product rule?

May 13, 2017

$f ' \left(x\right) = - 20 x \sin \left(x\right) - 5 \sin \left(x\right) - 10 {x}^{2} \cos \left(x\right) - 5 x \cos \left(x\right) + 5 \cos \left(x\right)$

Explanation:

Product Rule:
$\left[f \left(x\right) g \left(x\right)\right] ' = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$
Set $f \left(x\right) = 5 \sin \left(x\right)$ and $g \left(x\right) = \left(1 - 2 x\right) \left(x + 1\right)$

(1)
$\frac{d}{\mathrm{dx}} \left[5 \sin \left(x\right)\right] = 5 \cos \left(x\right)$

(2)
$\frac{d}{\mathrm{dx}} \left[\left(1 - 2 x\right) \left(x + 1\right)\right] = \frac{d}{\mathrm{dx}} \left[- 2 {x}^{2} - x + 1\right] = - 4 x - 1$

Putting it all together
$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left[5 \sin \left(x\right) \left(1 - 2 x\right) \left(x + 1\right)\right] - \frac{d}{\mathrm{dx}} \left[3\right]$
$f ' \left(x\right) = \left(5 \sin \left(x\right) \left(- 4 x - 1\right)\right) + 5 \cos \left(x\right) \left(- 2 {x}^{2} - x + 1\right) - 0$
$f ' \left(x\right) = - 20 x \sin \left(x\right) - 5 \sin \left(x\right) - 10 {x}^{2} \cos \left(x\right) - 5 x \cos \left(x\right) + 5 \cos \left(x\right)$