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To solve for \( h'(0) \), where \( h(x) = \frac{f(x)}{g(x)} \), we will use the quotient rule for derivatives. 

The quotient rule states:

\[
h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\]

We are asked to find \( h'(0) \), which means evaluating this expression at \( x = 0 \).

From the table:
- \( f(0) = 3 \)
- \( f'(0) = 4 \)
- \( g(0) = 2 \)
- \( g'(0) = \pi \)

Substitute these values into the quotient rule formula:

\[
h'(0) = \frac{f'(0)g(0) - f(0)g'(0)}{(g(0))^2}
\]

Now, plug in the values:

\[
h'(0) = \frac{(4)(2) - (3)(\pi)}{(2)^2}
\]

Simplify:

\[
h'(0) = \frac{8 - 3\pi}{4}
\]

Thus, the value of \( h'(0) \) is:

\[
h'(0) = \frac{8 - 3\pi}{4}
\]

Would you like further clarification on this, or any additional details?

Here are five related questions for practice:
1. How do you apply the product rule in derivatives?
2. If \( h(x) = f(x) \cdot g(x) \), how would you calculate \( h'(x) \)?
3. What are the key differences between the product rule and the quotient rule?
4. How does the chain rule apply when differentiating composite functions?
5. Can you calculate the second derivative \( h''(x) \) using the quotient rule?

**Tip:** Always ensure the denominator \( g(x) \) is non-zero when applying the quotient rule to avoid undefined expressions.

If h(x) = f(x) / g(x), what is the value of h'(0)?

Learn how to find the derivative h'(0) using the quotient rule for the given functions f(x) and g(x). This calculus problem covers the quotient rule and its application in finding the derivative of a quotient of two differentiable functions.

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