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Let's solve the function \( y(x) = 8\sqrt{x} \) by finding its derivative and then evaluating the derivative at the specified \( x \)-coordinates.

### Step 1: Rewrite the function
The function \( y(x) = 8\sqrt{x} \) can be rewritten as:
\[
y(x) = 8x^{\frac{1}{2}}
\]

### Step 2: Find the derivative
Using the power rule:
\[
\frac{dy}{dx} = 8 \times \frac{1}{2}x^{-\frac{1}{2}} = 4x^{-\frac{1}{2}} = \frac{4}{\sqrt{x}}
\]

### Step 3: Evaluate the derivative at the given \( x \)-coordinates

Now, we'll evaluate the derivative at the following \( x \)-coordinates:
- \( x_1 = 3.01 \)
- \( x_2 = 5.04 \)
- \( x_3 = 10.77 \)

The derivative expression to evaluate is:
\[
\frac{dy}{dx} = \frac{4}{\sqrt{x}}
\]

Let's compute these values now.Here are the computed derivatives for the given \( x \)-coordinates:

1. \( \frac{dy}{dx} \) at \( x_1 = 3.01 \) is approximately **2.3056**.
2. \( \frac{dy}{dx} \) at \( x_2 = 5.04 \) is approximately **1.7817**.
3. \( \frac{dy}{dx} \) at \( x_3 = 10.77 \) is approximately **1.2189**.

These values can be entered into the table. Would you like further details or have any questions?

### Additional Related Questions:
1. How does the derivative of a square root function generally behave as \( x \) increases?
2. What is the effect of the coefficient 8 on the derivative of this function?
3. How does this derivative compare to the derivative of \( y(x) = \sqrt{x} \) without any constant multiplication?
4. How would the derivative change if the function was \( y(x) = 8x^{3/2} \)?
5. What does the derivative tell us about the slope of the tangent at these specific \( x \)-values?

**Tip:** When dealing with square root functions, it's useful to remember that the derivative will tend to infinity as \( x \) approaches zero, indicating steepness in the graph near the origin.

Learn how to solve the function y(x) = 8√x by finding its derivative using the power rule. Explore the behavior of the derivative at specific x-values and understand its implications. Suitable for college-level mathematics enthusiasts.

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