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We are asked to find the derivative of the function:

\[
y = \frac{x^6}{\sqrt{x} + 4x^9}
\]

using the quotient rule. The quotient rule states that if \( y = \frac{f(x)}{g(x)} \), then the derivative \( y' \) is given by:

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

Now, let's identify \( f(x) \) and \( g(x) \) based on the given function.

- \( f(x) = x^6 \)
- \( g(x) = \sqrt{x} + 4x^9 = x^{1/2} + 4x^9 \)

### Step 1: Find \( f'(x) \)
\[
f'(x) = \frac{d}{dx} (x^6) = 6x^5
\]

### Step 2: Find \( g'(x) \)
\[
g'(x) = \frac{d}{dx} \left( x^{1/2} + 4x^9 \right) = \frac{1}{2}x^{-1/2} + 36x^8
\]

### Step 3: Apply the quotient rule
Now, using the quotient rule:

\[
y' = \frac{(6x^5)(x^{1/2} + 4x^9) - (x^6)\left(\frac{1}{2}x^{-1/2} + 36x^8 \right)}{(x^{1/2} + 4x^9)^2}
\]

This is the derivative of the given function using the quotient rule.

Would you like to proceed with simplifying the derivative further or leave it in this form? Let me know if you have any questions!

### Related Questions:
1. What are some common mistakes to avoid when using the quotient rule?
2. How do you handle functions with square roots in derivatives?
3. Can you apply the product rule in this situation instead of the quotient rule?
4. How does simplifying the numerator of a quotient rule derivative help in understanding the final expression?
5. What is the difference between the chain rule and the quotient rule?

**Tip:** Always remember to square the denominator when applying the quotient rule.

Calculate the derivative of the function y = x^6 / (sqrt(x) + 4x^9) using the quotient rule.

Learn how to calculate the derivative of the function y = x^6 / (sqrt(x) + 4x^9) using the quotient rule. This problem involves applying the quotient rule, power rule, and simplifying the expression. Step-by-step solution provided. Suitable for advanced calculus students in Grades 11-12.

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