To complete the calculation of $h'(1)$ for the function $h(x) = f(2x)(g(x))^2$, let's break down the steps more carefully.

Step-by-Step Solution

Given:

• $h(x) = f(2x) \cdot (g(x))^2$
• $f(1) = 3$
• Values of $g(x)$ and $g'(x)$:

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

Differentiation

To find $h'(x)$, we use the product rule:

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

Evaluate $h'(1)$:

1. Calculate $f'(2)$ and $f(2)$:

   We need the values of $f'(2)$ and $f(2)$ from the graph of $f'(x)$. Since we cannot directly access this graph from the text provided, let's proceed with placeholders:

   Let $f'(2) = a$ and $f(2) = b$.

2. Substitute the known values:

   At $x = 1$: $h'(1) = f'(2) \cdot 2 \cdot (g(1))^2 + f(2) \cdot 2 \cdot g(1) \cdot g'(1)$

   Substitute $g(1) = 35$, $g'(1) = -2$, and the placeholders $f'(2) = a$, $f(2) = b$:

   $h'(1) = a \cdot 2 \cdot 35^2 + b \cdot 2 \cdot 35 \cdot (-2)$

3. Simplify the expression:

   Calculate $35^2 = 1225$:

   $h'(1) = 2a \cdot 1225 + b \cdot 2 \cdot 35 \cdot (-2)$ $h'(1) = 2450a - 140b$

Conclusion:

The exact value of $h'(1)$ is:

$h'(1) = 2450a - 140b$

where $a = f'(2)$ and $b = f(2)$. We need the values of $f(2)$ and $f'(2)$ from the graph of $f'(x)$ to complete the numerical calculation.

If you have access to the graph of $f'(x)$ or any additional information about the function values, please provide it so that we can finalize the result.

Related Questions

1. What is the derivative of a product of two functions?
2. How can we interpret the derivative values graphically?
3. What are the common differentiation rules in calculus?
4. How does the chain rule work when differentiating composite functions?
5. How do we handle higher-order derivatives in multivariable calculus?

Tip

Always ensure you have all necessary values or graphs to fully complete a problem involving differentiation, especially when external data like a graph is involved.