1. The problem is to find the derivative of the function $\sqrt{u}$ with respect to $u$. 2. Recall the formula for the derivative of a power function: if $f(u) = u^n$, then $f'(u) = n u^{n-1}$. 3. The square root function can be rewritten as a power function: $\sqrt{u} = u^{\frac{1}{2}}$. 4. Applying the power rule for derivatives: $$\frac{d}{du} u^{\frac{1}{2}} = \frac{1}{2} u^{\frac{1}{2} - 1} = \frac{1}{2} u^{-\frac{1}{2}}$$ 5. Simplify the expression: $$\frac{1}{2} u^{-\frac{1}{2}} = \frac{1}{2 \sqrt{u}}$$ 6. Therefore, the derivative of $\sqrt{u}$ is: $$\boxed{\frac{d}{du} \sqrt{u} = \frac{1}{2 \sqrt{u}}}$$ This means the rate of change of the square root of $u$ with respect to $u$ is $\frac{1}{2 \sqrt{u}}$.