Derivative of \(\cos \left(\sqrt{x}\right)\) is :

\(\text{ We need to differentiate  }\cos (\sqrt{x})\text{  with respect to }x\text{. }\)

Step 1: Apply Chain Rule

Let: \(y=\cos \left(\sqrt{x}\right)\)

\(\text{ Define }u=\sqrt{x}\Rightarrow {x}^{1/2}\text{, so that: }\\ y=\cos \left(u\right)\)

Now, differentiate both sides:

\(\frac{dy}{dx}=\frac{d}{du}\cos (u)⋅\frac{du}{dx}\)

Step 2: Compute the Derivatives

\(\frac{d(\cos u)}{du}=-\sin (u)\\ \frac{du}{dx}=\frac{d\left({x}^{(\frac{1}{2})}\right)}{dx}\)


Step 3: Multiply the Terms

\(\frac{dy}{dx}=-\frac{\sin \sqrt{x}}{2\sqrt{x}}\)