Differentiate the function with respect to $x$: $\cos (\sqrt{x})$

Let $f(x) = \cos (\sqrt{x})$.
Using the chain rule,we differentiate the outer function $\cos(u)$ where $u = \sqrt{x}$,and then multiply by the derivative of the inner function $u = \sqrt{x}$.
$\frac{d}{dx} [\cos (\sqrt{x})] = -\sin (\sqrt{x}) \cdot \frac{d}{dx} (\sqrt{x})$
Since $\frac{d}{dx} (\sqrt{x}) = \frac{d}{dx} (x^{1/2}) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}}$,we substitute this back into the expression:
$= -\sin (\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}$
$= -\frac{\sin (\sqrt{x})}{2\sqrt{x}}$