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Let's solve this problem step by step.

We are given the function:
\[
g(x) = 2\sqrt{x}
\]
We are asked to find the slope of the tangent line at \( x = 9 \). 

### (a) Finding the Slope of the Tangent Line
To find the slope of the tangent line, we need to compute the derivative of \( g(x) \), which represents the slope of the tangent line at any point \( x \). Let's differentiate \( g(x) = 2\sqrt{x} \) using the chain rule.

1. Rewrite \( g(x) \) as:
   \[
   g(x) = 2x^{1/2}
   \]

2. Now differentiate:
   \[
   g'(x) = 2 \cdot \frac{1}{2}x^{-1/2} = \frac{1}{\sqrt{x}}
   \]

3. Evaluate the derivative at \( x = 9 \):
   \[
   g'(9) = \frac{1}{\sqrt{9}} = \frac{1}{3}
   \]

So, the slope of the tangent line at \( x = 9 \) is \( \frac{1}{3} \).

Would you like the details for part (b) or more explanations on the derivative?

Here are 5 related questions to explore:
1. How do we interpret the slope of a tangent line geometrically?
2. What are the common rules for differentiating power functions?
3. How do you find the equation of a tangent line after knowing the slope?
4. How does the chain rule apply to other composite functions?
5. How does the slope of a tangent line vary with different functions?

Tip: Always check if your function can be rewritten to apply simpler derivative rules, like converting roots into fractional exponents!

For the given function, (a) find the slope of the tangent line to the graph at the given point; (b) find the equation of the tangent line. Given g(x) = 2√x at x = 9.

Calculate the slope of the tangent line to the function g(x) = 2√x at x = 9. This calculus problem involves differentiation using the chain rule and finding the slope of the tangent line. Suitable for students in Grades 10-12.

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