To find the derivative of the function $y = \ln\sqrt{x} + 3$, follow these steps:

Step 1: Simplify the expression

We know that $\sqrt{x} = x^{1/2}$. Hence, the function can be rewritten as: $y = \ln x^{1/2} + 3$

Using the logarithmic property $\ln a^b = b \ln a$, this becomes: $y = \frac{1}{2} \ln x + 3$

Step 2: Differentiate term by term

The derivative of $y = \frac{1}{2} \ln x + 3$ with respect to $x$ is: $\frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{x} + 0$

Thus, the derivative is: $\boxed{\frac{dy}{dx} = \frac{1}{2x}}$

---

Do you want further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the derivative of $y = \ln(x^2 + 1) + 5$?
2. How would the derivative change if $y = \ln x^{1/3} + 3$?
3. What is the chain rule and how does it apply to differentiating logarithmic functions?
4. How do you differentiate $y = \ln(\sin x)$?
5. How can you integrate $\frac{1}{2x}$ to recover the original function?

Tip: Always simplify logarithmic expressions using properties like $\ln a^b = b \ln a$ before differentiating—they make the process much easier!