To solve the problem, we need to find the derivative of $h(x) = (f(x))^2$ and then evaluate it at $x = 2$.

Step 1: Differentiate $h(x)$

The function $h(x) = (f(x))^2$ is a composition of functions, so we need to apply the chain rule to differentiate it:

$h'(x) = 2f(x) \cdot f'(x)$

Step 2: Substitute the given values

Now, we substitute $x = 2$, $f(2) = 4$, and $f'(2) = 5$:

$h'(2) = 2 \cdot f(2) \cdot f'(2)$

Substitute the given values:

$h'(2) = 2 \cdot 4 \cdot 5$

Step 3: Calculate the value

$h'(2) = 2 \cdot 20 = 40$

Final Answer

The value of $h'(2)$ is 40.

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Related Questions

1. What is the chain rule in calculus, and how is it applied to functions?
2. How do you find the derivative of a composite function?
3. What is the significance of the derivative at a specific point?
4. Can you solve for $h'(x)$ if $h(x) = f(x)g(x)$ and both $f(x)$ and $g(x)$ are given?
5. How does the product rule differ from the chain rule?

Tip

When differentiating a function that involves another function raised to a power, remember to use the chain rule by differentiating the outer function first and then multiplying by the derivative of the inner function.