# How do you find the derivative of y=2cosxsinx

Feb 25, 2015

$2 \cos \left(2 x\right)$

#### Explanation:

Since $y = 2 \cos \left(x\right) \sin \left(x\right)$ is the same as
$y = \sin \left(2 x\right)$

and
$\frac{d \sin \left(\theta\right)}{d \theta} = \cos \left(\theta\right)$

Let $g \left(a\right) = \sin \left(a\right)$ and $h \left(b\right) = 2 b$
(So $y = g \left(h \left(x\right)\right)$)

By the chain rule:
$\frac{d y}{\mathrm{dx}} = \frac{d g \left(h \left(x\right)\right)}{d h \left(x\right)} \cdot \frac{d h \left(x\right)}{\mathrm{dx}}$

$\textcolor{w h i t e}{\left(\mathrm{dy}\right) \left(\mathrm{dx}\right)} = \frac{d \left(\sin \left(2 x\right)\right)}{d \left(2 x\right)} \cdot \frac{d \left(2 x\right)}{\mathrm{dx}}$

$\textcolor{w h i t e}{\left(\mathrm{dy}\right) \left(\mathrm{dx}\right)} = \cos \left(2 x\right) \cdot 2$
or
$\textcolor{w h i t e}{\left(\mathrm{dy}\right) \left(\mathrm{dx}\right)} = 2 \cos \left(2 x\right)$