Find the derivative of the function.

*y* =

3

√*x* +6

To find the derivative of the function \( y = 3\sqrt{x} + 6 \), we'll differentiate it with respect to \( x \).

### Step 1: Express the square root in exponent form
The function can be rewritten as:
\[
y = 3x^{\frac{1}{2}} + 6
\]

### Step 2: Differentiate each term with respect to \( x \)
- The derivative of \( 3x^{\frac{1}{2}} \) with respect to \( x \):
  \[
  \frac{d}{dx}\left(3x^{\frac{1}{2}}\right) = 3 \cdot \frac{1}{2}x^{\frac{1}{2} - 1} = \frac{3}{2}x^{-\frac{1}{2}}
  \]
- The derivative of the constant \( 6 \) is \( 0 \).

### Step 3: Combine the results
Thus, the derivative of the function is:
\[
\frac{dy}{dx} = \frac{3}{2}x^{-\frac{1}{2}} = \frac{3}{2\sqrt{x}}
\]

So, the derivative of \( y = 3\sqrt{x} + 6 \) is:
\[
\frac{dy}{dx} = \frac{3}{2\sqrt{x}}
\]

Would you like to go over more details or have any questions? 

Here are 5 related questions:
1. How do you find the derivative of a general power function \( x^n \)?
2. What is the derivative of \( y = \sqrt{x} \)?
3. How do you differentiate a function with multiple terms?
4. What happens to the derivative if we multiply the function by a constant?
5. How does the chain rule apply to more complex root functions?

**Tip:** When differentiating square roots, it's often easier to first rewrite the root as a fractional exponent.

Learn how to find the derivative of the function y = 3√x + 6 using the power rule of differentiation. Step-by-step solution provided. Suitable for advanced high school students and college-level calculus learners.

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