https://file.aiquickdraw.com/mathbot/file/172533263294477v96wlc.pdf

The PDF appears to contain a mathematics assignment, with the first question involving the calculation of the derivative \( h'(1) \) for a function \( h(x) = f(2x)(g(x))^2 \), given certain values and the graph of \( f'(x) \). Here is a detailed breakdown of the information provided:

### Question 1

1. **Function Definition:**
   - \( h(x) = f(2x) \cdot (g(x))^2 \)
   - \( f(x) \) is a function with a known value \( f(1) = 3 \).
   - The graph of \( f'(x) \) is provided (though we cannot see the graph).

2. **Values Provided:**
   - \( g(x) \) and \( g'(x) \) are given for \( x = 0, 1, 2 \):

   | \(x\) | \(g(x)\) | \(g'(x)\) |
   |---|---|---|
   | 0 | 10 | 5  |
   | 1 | 35 | -2 |
   | 2 | 30 | 2  |

### Solution for \( h'(x) \)

To find \( h'(x) \), we apply the product rule:

\[
h'(x) = \frac{d}{dx} [f(2x) \cdot (g(x))^2] = f'(2x) \cdot 2 \cdot (g(x))^2 + f(2x) \cdot 2g(x)g'(x)
\]

At \( x = 1 \):

\[
h'(1) = f'(2) \cdot 2 \cdot (g(1))^2 + f(2) \cdot 2 \cdot g(1) \cdot g'(1)
\]

Using the provided values:

- \( g(1) = 35 \), \( g'(1) = -2 \)
- We need to know \( f(2) \) and \( f'(2) \). These may require the graph provided in the document.

Would you like me to proceed further with these calculations, or do you have access to the graph of \( f'(x) \) that could provide the necessary values?

Feel free to provide further instructions or upload additional information. 

### Related Questions

1. How do you apply the chain rule in calculus?
2. What is the significance of the product rule when differentiating functions?
3. How can we find the derivative of a product of three functions?
4. How is the value of a derivative graphically interpreted?
5. What are the practical applications of finding derivatives in real-world problems?

### Tip
When dealing with composite functions, make sure to correctly identify and apply the chain rule and product rule to avoid errors in differentiation.

Learn how to calculate the derivative h'(1) using differentiation and the product rule. This problem involves a composite function h(x) involving functions f(x) and g(x), and requires knowledge of derivatives and their applications. Suitable for advanced high school or college-level mathematics.

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