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To solve the derivative of the given function:

\[
y = \frac{x^4 + 6}{x^2}
\]

First, simplify the expression by dividing each term in the numerator by \(x^2\):

\[
y = \frac{x^4}{x^2} + \frac{6}{x^2} = x^2 + 6x^{-2}
\]

Now, we differentiate each term with respect to \(x\):

1. The derivative of \(x^2\) is \(2x\).
2. The derivative of \(6x^{-2}\) is \(6 \cdot (-2) x^{-3} = -12x^{-3}\).

Thus, the derivative \(\frac{dy}{dx}\) is:

\[
\frac{dy}{dx} = 2x - \frac{12}{x^3}
\]

Would you like further details on any of these steps? 

Here are 5 related questions to expand your understanding:
1. How do you differentiate a quotient without simplifying first?
2. What are the rules for differentiating powers of \(x\)?
3. How do you handle derivatives of negative exponents?
4. Can you apply the chain rule to more complex functions?
5. How would the result differ if the function were \(\frac{6 + x^4}{x}\) instead?

**Tip:** Simplifying expressions before differentiating often makes the process easier and less error-prone!

Find the derivative of the function: y = (x^4 + 6) / x^2

Learn how to differentiate the function y = (x^4 + 6) / x^2 by simplifying the expression and applying the power rule. This step-by-step guide explains the key concepts and methods in differentiation, suitable for students in Grades 11-12.

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