Differentiate.

*y* = (*x*2 − 9)

*x*

*y'* =

To find the derivative of \( y = \frac{x^2 - 9}{x} \), we can first simplify the expression and then differentiate:

### Step 1: Rewrite \( y \) in terms of powers of \( x \)
Rewrite \( y \) as:
\[
y = \frac{x^2}{x} - \frac{9}{x} = x - 9x^{-1}
\]

### Step 2: Differentiate \( y \) with respect to \( x \)
Now, apply the power rule to each term:
1. The derivative of \( x \) is \( 1 \).
2. The derivative of \( -9x^{-1} \) is \( 9x^{-2} \) (using the power rule \( \frac{d}{dx} x^{-n} = -nx^{-n-1} \)).

So,
\[
y' = 1 + 9x^{-2}
\]

### Final Answer:
\[
y' = 1 + \frac{9}{x^2}
\]

Would you like further details or have any questions?

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**Related Questions:**
1. How do you find the derivative of a quotient using the quotient rule?
2. What are other ways to simplify expressions before differentiating?
3. How does the power rule apply to negative exponents?
4. What would the derivative look like if we had a more complex denominator?
5. How would this derivative change if there were an additional constant term?

**Tip:** Simplifying an expression before differentiating can make the process easier, especially for quotients or products.

Find the derivative of y = (x^2 - 9) / x with a step-by-step solution. This calculus problem covers concepts in differentiation, including simplifying expressions and applying the power rule. Suitable for high school calculus students (Grades 11-12).

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