What is the derivative of $x\sin x$? Ok so I know the answer of $\frac{d}{dx}x\sin(x) = \sin(x)+ x\cos(x)$...but how exactly do you get there? I know $\frac{d}{dx} \sin{x} = \cos{x}$. But where does the additional $\sin(x)$ (in the answer) come in? 

You are probably expected to do this by the product rule, which says:
$(v \cdot w)'=vw'+v'w$
Setting $v=x$ and $w=\sin x$ (it doesn't matter which is which)
$v'=1$ and $w'=\cos x$ . Substituting into the formula:
$(v \cdot w)'=x \cos x+(1) \cdot \sin x$ 
$x \cos x+ \sin x$ 
which is your answer.
