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give me the numerical value of the derivative of h(0)

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### Given:
\[
h(x) = 3f(x) - 2g(x) - 5 \cos(x) - 3
\]
You are asked to find \( h'(0) \) given:
- \( f(0) = 3 \), \( g'(0) = 7 \)
- \( f \) and \( g \) are differentiable functions.

We start by differentiating \( h(x) \):
\[
h'(x) = 3f'(x) - 2g'(x) - 5(-\sin(x))
\]
Now, substitute \( x = 0 \):
\[
h'(0) = 3f'(0) - 2g'(0) + 5\sin(0)
\]
Since \( \sin(0) = 0 \), this simplifies to:
\[
h'(0) = 3f'(0) - 2g'(0)
\]
Substitute \( g'(0) = 7 \):
\[
h'(0) = 3f'(0) - 14
\]

To get the exact value of \( h'(0) \), we would need \( f'(0) \), which isn't given explicitly in the problem. However, this is the expression for \( h'(0) \).

Would you like to proceed with additional details or calculations? Here are five related questions to deepen your understanding:

1. What is the significance of knowing both \( f'(0) \) and \( g'(0) \) in determining \( h'(0) \)?
2. How would the derivative of \( h(x) \) change if \( f(x) \) or \( g(x) \) were not differentiable?
3. Why does the sine function \( \sin(0) = 0 \) simplify the problem?
4. How would the problem change if the constants in front of \( f(x) \) and \( g(x) \) were different?
5. Can the problem be extended to higher-order derivatives of \( h(x) \)?

### Tip:
When differentiating composite functions, remember to apply the chain rule if any of the functions have more complex structures (like compositions).

Give me the numerical value of the derivative of h(0) given h(x) = 3f(x) - 2g(x) - 5cos(x) - 3 and f'(0) = 3, g'(0) = 7.

Calculate the derivative of h(x) = 3f(x) - 2g(x) - 5cos(x) - 3 at x=0. In this step-by-step solution, we apply the rules of differentiation, given that f'(0) = 3 and g'(0) = 7, and simplify using trigonometric identities.

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