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We are given that \( f \) and \( g \) are differentiable functions, with \( f'(0) = 3 \) and \( g'(0) = 7 \). We are also given a function \( h(x) \) defined as:

\[
h(x) = 3f(x) - 2g(x) - 5\cos(x) - 3
\]

We need to find the value of \( h'(0) \).

### Step-by-step solution:

1. **Differentiate \( h(x) \):**

   To find \( h'(x) \), we differentiate \( h(x) \) term by term:

   \[
   h(x) = 3f(x) - 2g(x) - 5\cos(x) - 3
   \]

   - The derivative of \( 3f(x) \) is \( 3f'(x) \).
   - The derivative of \( -2g(x) \) is \( -2g'(x) \).
   - The derivative of \( -5\cos(x) \) is \( 5\sin(x) \).
   - The derivative of the constant \( -3 \) is 0.

   So, the derivative \( h'(x) \) is:

   \[
   h'(x) = 3f'(x) - 2g'(x) + 5\sin(x)
   \]

2. **Evaluate \( h'(0) \):**

   To find \( h'(0) \), we substitute \( x = 0 \) into the derivative:

   \[
   h'(0) = 3f'(0) - 2g'(0) + 5\sin(0)
   \]

   - \( f'(0) = 3 \)
   - \( g'(0) = 7 \)
   - \( \sin(0) = 0 \)

   Substituting these values:

   \[
   h'(0) = 3(3) - 2(7) + 5(0)
   \]

   Simplifying:

   \[
   h'(0) = 9 - 14 + 0 = -5
   \]

Thus, the value of \( h'(0) \) is \( \boxed{-5} \).

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Would you like more details on this solution, or do you have any additional questions?

Here are 5 related questions to explore further:

1. How do you apply the chain rule to differentiate composite functions?
2. What is the significance of differentiability at a point?
3. How would the result change if \( h(x) \) had a \( \sin(x) \) term instead of \( \cos(x) \)?
4. What is the relationship between a function’s derivative and the tangent line at a point?
5. How do you compute higher-order derivatives for more complex functions?

**Tip:** Always check if the terms you are differentiating are constants, as their derivatives will simplify the expression significantly!

Let f and g be differentiable functions such that f'(0) = 3 and g'(0) = 7. If h(x) = 3f(x) - 2g(x) - 5cos(x) - 3, what is the value of h'(0)?

Find the derivative h'(0) for the function h(x) = 3f(x) - 2g(x) - 5cos(x) given f'(0) = 3 and g'(0) = 7. Learn step-by-step how to differentiate and apply the chain rule for differentiable functions.

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