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.