Need help simplifying a differentiated product

Question

$\frac{d}{dx}[(2x)(\sqrt x)(\sin x)]$

This was the product differentiated:

$2 \sqrt x \sin x + \frac{-2x \sin x}{2\sqrt x} + 2x \sqrt x \cos x$

My simplification was:

$2\sqrt x \sin x + (-\sqrt x \sin x) + 2x \sqrt x \cos x$

However, this was wrong as the correct answer is:

$3\sqrt x \sin x + 2x \sqrt x \cos x$

Can someone explain where I went wrong with the simplification?? Thanks

Answer 1

$$\frac{d}{dx} \sqrt{x} = \frac1{2\sqrt{x}}$$

There is no negative sign. Square root is an increasing function.

Answer 2

You do not need a negative sign in $$-1/2 \sqrt x $$

It is much easier if you multiply

$$x\sqrt x $$ to get $$x^{3/2}$$ and differentiate

$$ ( 2x^{3/2} \sin x)'=3x^{1/2}\sin x +2x^{3/2}\cos x$$