before I ask for anything I must admit I'm working hard to understand this beautiful subject. Thanks in advance. I want to get a derivative of: $\cos(x)\sin(x)$ The solution is the following:
$$ \frac d{dx}(\cos(x) \sin(x)) = \cos(2 x)$$
Where should I start?
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$\begingroup$One way is to recall the identity $\sin 2x=2\sin x\cos x$, which can be rewritten as $$\sin x\cos x=\frac{1}{2}\sin 2x.$$ Now differentiate, not forgetting to use the Chain Rule.
An alternative is to differentiate $\cos x\sin x$ immediately, using the Product Rule. We get $(\cos x)(\cos x)+(-\sin x)(\sin x)=\cos^2 x-\sin^2 x$. Then recall the identity $\cos 2x=\cos^2 x-\sin^2 x$.
$\endgroup$ 2 $\begingroup$Do you know the rules for taking the derivatives of products?
$$\frac d{dx}(uv) = u'v + uv'$$
Set $u = \cos(x)$ and $v = \sin(x)$ and find their derivatives using this rule.
Then you will need to use a Double-Angle Formula to get your desired result.
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