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.