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

Feb 3, 2017

$f ' \left(x\right) = \left(3 {x}^{2} + 6 x\right) \cos x + \left(6 x + 6\right) \sin x$

Explanation:

$\text{Given "f(x)=g(x)h(x)" then}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{f ' \left(x\right) = g \left(x\right) h ' \left(x\right) + h \left(x\right) g ' \left(x\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}} \leftarrow \text{ product rule}$

We can express f(x) as the product of 2 functions.

$\Rightarrow f \left(x\right) = \left(3 {x}^{2} + 6 x\right) \sin x$

$\text{here } g \left(x\right) = 3 {x}^{2} + 6 x \Rightarrow g ' \left(x\right) = 6 x + 6$

$\text{and } h \left(x\right) = \sin x \Rightarrow h ' \left(x\right) = \cos x$

$\Rightarrow f ' \left(x\right) = \left(3 {x}^{2} + 6 x\right) \cos x + \left(6 x + 6\right) \sin x$