https://file.aiquickdraw.com/mathbot/file/1727274903216f7nb63ye.jpg

We are given the function:

\[
g(u) = \sqrt{3u} + \sqrt{2u}
\]

To find \( g'(u) \), we need to differentiate this expression with respect to \( u \).

### Step-by-step solution:

1. Rewrite the square roots as exponents:
   \[
   g(u) = (3u)^{1/2} + (2u)^{1/2}
   \]

2. Apply the chain rule to differentiate each term:
   - For \( (3u)^{1/2} \), use the chain rule:
     \[
     \frac{d}{du} \left( (3u)^{1/2} \right) = \frac{1}{2} (3u)^{-1/2} \cdot 3 = \frac{3}{2\sqrt{3u}}
     \]
   - For \( (2u)^{1/2} \), similarly:
     \[
     \frac{d}{du} \left( (2u)^{1/2} \right) = \frac{1}{2} (2u)^{-1/2} \cdot 2 = \frac{1}{\sqrt{2u}}
     \]

3. Combine the results:
   \[
   g'(u) = \frac{3}{2\sqrt{3u}} + \frac{1}{\sqrt{2u}}
   \]

Thus, the derivative of \( g(u) \) is:

\[
g'(u) = \frac{3}{2\sqrt{3u}} + \frac{1}{\sqrt{2u}}
\]

Let me know if you would like further clarification or have any questions!

Here are some related questions to expand your understanding:
1. How do you apply the chain rule to more complicated functions?
2. What happens if you differentiate a product of functions involving square roots?
3. How would you integrate the function \( g(u) \)?
4. What is the difference between implicit and explicit differentiation?
5. How can the power rule be used to differentiate terms with fractional exponents?

**Tip:** Always check if functions can be simplified before differentiation—it often makes the process easier!

Differentiate the function g(u) = √(3u) + √(2u).

Learn how to differentiate the function g(u) = √(3u) + √(2u) using the chain rule and square root derivatives. A step-by-step solution is provided, suitable for students in Grades 11-12 studying advanced calculus.

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