Find the derivative of the function.
yequalsStartFraction x Superscript 4 Baseline plus 7 Over x squared EndFraction
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Part 1
Simplify y by dividing each term in the numerator by x squared.

We are given the function:

\[
y = \frac{x^4 + 7}{x^2}
\]

To simplify this, divide each term in the numerator by \(x^2\):

\[
y = \frac{x^4}{x^2} + \frac{7}{x^2}
\]

Simplifying each term:

\[
y = x^{4-2} + 7x^{-2}
\]

\[
y = x^2 + 7x^{-2}
\]

Now that the function is simplified, we can proceed to find the derivative.

Would you like to continue with finding the derivative?

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### Follow-up questions:
1. What are the rules for simplifying fractions involving exponents?
2. How does dividing a polynomial by a monomial simplify a function?
3. What is the power rule for derivatives, and how does it apply here?
4. How would you approach finding the derivative of a more complex function?
5. What are some real-world applications of derivatives?

**Tip:** Simplifying functions before differentiation can make finding derivatives much easier.

Find the derivative of the function y = (x^4 + 7) / x^2. Simplify y by dividing each term in the numerator by x^2.

This detailed guide shows how to find the derivative of the function y = (x^4 + 7) / x^2 by first simplifying the expression. Key concepts include algebraic simplifications and the application of the power rule for derivatives. Suitable for high school students in Grades 10-12.

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