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To find the derivative of \( h(x) = -\frac{7x}{\sqrt{9x^3 + 4}} \), we'll use the quotient rule. The quotient rule states that if we have a function of the form \( \frac{f(x)}{g(x)} \), its derivative is given by:

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

In this case, let:

- \( f(x) = -7x \)
- \( g(x) = \sqrt{9x^3 + 4} \)

We need to find \( f'(x) \) and \( g'(x) \) separately.

1. **Derivative of \( f(x) \):**

\[
f'(x) = -7
\]

2. **Derivative of \( g(x) \):**

To find \( g'(x) \), we use the chain rule. Recall that \( g(x) = (9x^3 + 4)^{1/2} \). The derivative of \( g(x) \) is:

\[
g'(x) = \frac{1}{2}(9x^3 + 4)^{-1/2} \cdot (27x^2)
\]

Simplifying:

\[
g'(x) = \frac{27x^2}{2\sqrt{9x^3 + 4}}
\]

3. **Applying the Quotient Rule:**

Now, applying the quotient rule formula:

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

Substitute \( f'(x) = -7 \), \( f(x) = -7x \), \( g(x) = \sqrt{9x^3 + 4} \), and \( g'(x) = \frac{27x^2}{2\sqrt{9x^3 + 4}} \):

\[
h'(x) = \frac{(-7)\sqrt{9x^3 + 4} - (-7x)\left(\frac{27x^2}{2\sqrt{9x^3 + 4}}\right)}{(9x^3 + 4)}
\]

Simplify further:

\[
h'(x) = \frac{-7\sqrt{9x^3 + 4} + \frac{189x^3}{2\sqrt{9x^3 + 4}}}{9x^3 + 4}
\]

Combine terms over the common denominator \( 2\sqrt{9x^3 + 4} \):

\[
h'(x) = \frac{-14(9x^3 + 4) + 189x^3}{2(9x^3 + 4)\sqrt{9x^3 + 4}}
\]

Simplify further:

\[
h'(x) = \frac{189x^3 - 14(9x^3 + 4)}{2(9x^3 + 4)\sqrt{9x^3 + 4}}
\]

Finally, simplifying inside the numerator:

\[
h'(x) = \frac{189x^3 - 126x^3 - 56}{2(9x^3 + 4)\sqrt{9x^3 + 4}}
\]

\[
h'(x) = \frac{63x^3 - 56}{2(9x^3 + 4)\sqrt{9x^3 + 4}}
\]

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

\[
h'(x) = \frac{63x^3 - 56}{2(9x^3 + 4)\sqrt{9x^3 + 4}}
\]

Let me know if you have any questions or need further clarification!

### Related Questions:
1. How do you find the derivative of a quotient using the quotient rule?
2. What is the chain rule, and how is it used in differentiating composite functions?
3. How can you simplify complex derivatives to a more concise form?
4. Why is it important to simplify the derivative fully before using it in further calculations?
5. What are the typical mistakes to avoid when using the quotient rule?

### Tip:
Always double-check your derivatives, especially when using the quotient rule, as small sign errors can significantly impact the final result.

Find the derivative of h(x) = - (7x) / sqrt(9x^3 + 4)

Learn how to find the derivative of h(x) = -7x / sqrt(9x^3 + 4) using the quotient rule and the chain rule. The solution includes step-by-step differentiation and simplification, suitable for undergraduate calculus or advanced high school students.

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