Solution assignment 03 Product and Quotient rule

Differentiate:

$y=2\sin(x)\cos(x)$

The product rule yields:

$y'=2[\cos(x)\cos(x)+\sin(x)\cdot-\sin(x)]=2[\cos^2(x)-\sin^2(x)]$ :

From trigonometry we know that we can rewrite the latter form as:

$y'=2\cos(2x)$

Because we know, also from trigonometry:

$2\sin(x)\cos(x)=\sin(2x)$

we have derived the following result. If:

$y=\sin(2x)$

then:

$y'=2\cos(2x)$

We would have found the same result if we would have applied the Chain rule.