Find g prime left parenthesis x right parenthesis for the given function.
g left parenthesis x right parenthesis equals StartRoot 3 x EndRoot
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Part 1
g prime left parenthesis x right parenthesis ​= 
  
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To find \( g'(x) \) for the function \( g(x) = \sqrt{3x} \), let's proceed with the following steps:

### Step 1: Rewrite the function using exponent notation.
We can rewrite the square root as a fractional exponent:

\[
g(x) = (3x)^{\frac{1}{2}}
\]

### Step 2: Apply the chain rule.
To differentiate, we'll use the chain rule. Let \( u = 3x \), so \( g(x) = u^{\frac{1}{2}} \).

First, differentiate the outer function \( u^{\frac{1}{2}} \):

\[
\frac{d}{du} \left( u^{\frac{1}{2}} \right) = \frac{1}{2} u^{-\frac{1}{2}}
\]

Now differentiate the inner function \( u = 3x \):

\[
\frac{d}{dx} (3x) = 3
\]

### Step 3: Combine the results.
Now multiply the derivatives from the chain rule:

\[
g'(x) = \frac{1}{2} (3x)^{-\frac{1}{2}} \cdot 3
\]

Simplify the expression:

\[
g'(x) = \frac{3}{2} (3x)^{-\frac{1}{2}}
\]

Finally, rewrite this in terms of a square root:

\[
g'(x) = \frac{3}{2\sqrt{3x}}
\]

### Final Answer:
\[
g'(x) = \frac{3}{2\sqrt{3x}}
\]

Would you like further clarification or details on any of these steps?

Here are 5 related questions you might find useful:
1. How do you differentiate \( \sqrt{ax} \) for any constant \( a \)?
2. What is the chain rule in calculus, and why is it important?
3. How can we differentiate functions with multiple variables?
4. Can you explain how to apply the power rule in differentiation?
5. What is the difference between using fractional exponents and square roots in calculus?

**Tip:** Remember, rewriting expressions using exponents makes differentiation easier when applying rules like the power rule!

Find g'(x) for the given function: g(x) = √(3x).

Learn how to find the derivative of g(x) = √(3x) using the chain rule in calculus. This step-by-step guide covers exponent rules, the chain rule, and the power rule, ideal for students in Grades 11-12 or introductory calculus courses.

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