Find the derivative of the function.

f(x) = 	(x^3 − 7x^2 + 1)/

x^2


To find the derivative of the function \( f(x) = \frac{x^3 - 7x^2 + 1}{x^2} \), we will simplify the function first and then differentiate. 

### Step 1: Simplify the Function

First, divide each term in the numerator by \( x^2 \) to simplify the function:

\[
f(x) = \frac{x^3}{x^2} - \frac{7x^2}{x^2} + \frac{1}{x^2}
\]

Simplifying each term:

\[
f(x) = x - 7 + x^{-2}
\]

### Step 2: Differentiate Each Term

Now, differentiate each term separately using the power rule, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

1. **Differentiate \(x\):**
   \[
   \frac{d}{dx}(x) = 1
   \]

2. **Differentiate \(-7\):**
   \[
   \frac{d}{dx}(-7) = 0
   \]

3. **Differentiate \(x^{-2}\):**
   \[
   \frac{d}{dx}(x^{-2}) = -2x^{-2-1} = -2x^{-3}
   \]

### Step 3: Combine the Derivatives

Combine the derivatives of each term to get the derivative of the entire function:

\[
f'(x) = 1 + 0 - 2x^{-3}
\]

Simplify the expression:

\[
f'(x) = 1 - 2x^{-3}
\]

Or, rewriting \( -2x^{-3} \) as \( -\frac{2}{x^3} \):

\[
f'(x) = 1 - \frac{2}{x^3}
\]

### Final Answer

The derivative of the function \( f(x) = \frac{x^3 - 7x^2 + 1}{x^2} \) is:

\[
f'(x) = 1 - \frac{2}{x^3}
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the second derivative of \(f(x) = \frac{x^3 - 7x^2 + 1}{x^2}\)?
2. How do you find critical points of \(f(x) = \frac{x^3 - 7x^2 + 1}{x^2}\)?
3. What happens to the derivative as \(x\) approaches 0?
4. How do you differentiate a quotient using the quotient rule?
5. What are the inflection points of the function \(f(x) = \frac{x^3 - 7x^2 + 1}{x^2}\)?

**Tip:** When differentiating functions with fractions, simplifying the expression first can often make the differentiation process easier.

Learn how to find the derivative of the function f(x) = (x^3 - 7x^2 + 1) / x^2 step-by-step. This problem involves differentiation using the power rule and quotient rule, suitable for advanced high school or college-level students.

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