Derivative of a product involving sqrt(x) and cos(x)

Rewrite $f(x) = x^{1/2}\cos(x)$, a product of $u(x) = x^{1/2}$ and $v(x) = \cos(x)$. By the power rule, $u'(x) = \frac{1}{2}x^{-1/2}$, and by the trigonometric derivative rule, $v'(x) = -\sin(x)$.

Apply the product rule $f'(x) = u'(x)v(x) + u(x)v'(x)$ to get $f'(x) = \frac{1}{2}x^{-1/2}\cos(x) - x^{1/2}\sin(x)$. This can also be written as $f'(x) = \dfrac{\cos(x)}{2\sqrt{x}} - \sqrt{x}\sin(x)$.