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The function given is:

\[
f(x) = \frac{7}{3x^3 + 4x^2}
\]

To find the derivative \(f'(x)\), we'll use the **quotient rule** because the function is in the form of a quotient. The quotient rule is given by:

\[
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
\]

For \(f(x)\), we have:
- \(u(x) = 7\), so \(u'(x) = 0\)
- \(v(x) = 3x^3 + 4x^2\), so we need to find \(v'(x)\)

### Step 1: Differentiate \(v(x)\)
Differentiate \(v(x) = 3x^3 + 4x^2\):

\[
v'(x) = \frac{d}{dx}(3x^3) + \frac{d}{dx}(4x^2)
\]

\[
v'(x) = 9x^2 + 8x
\]

### Step 2: Apply the Quotient Rule
Using the quotient rule:

\[
f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{(v(x))^2}
\]

Substituting the known values:

\[
f'(x) = \frac{0 \cdot (3x^3 + 4x^2) - 7 \cdot (9x^2 + 8x)}{(3x^3 + 4x^2)^2}
\]

Simplify the expression:

\[
f'(x) = \frac{-7(9x^2 + 8x)}{(3x^3 + 4x^2)^2}
\]

### Final Answer:
The derivative of the function is:

\[
f'(x) = \frac{-7(9x^2 + 8x)}{(3x^3 + 4x^2)^2}
\]

Would you like more details or have any questions?

### Related Questions
1. How does the quotient rule differ from the product rule in differentiation?
2. What are some common functions where the quotient rule is applied?
3. Can the derivative be simplified further?
4. How would you apply the chain rule to more complex functions?
5. What is the significance of the derivative in understanding the behavior of functions?

### Tip
When using the quotient rule, carefully identify and differentiate both the numerator and the denominator before substituting into the formula to avoid mistakes.

Learn how to find the derivative of the function f(x) = 7 / (3x^3 + 4x^2) using the quotient rule. This problem covers differentiation techniques, including the application of the quotient rule formula. Suitable for advanced high school students and calculus enthusiasts.

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