To find the derivative of the function $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$, we will differentiate each term separately. Let's go through this step-by-step.

Step 1: Rewrite the Function

First, rewrite the function to simplify the differentiation process: $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$ The term $\frac{1}{x^2}$ can be rewritten as $x^{-2}$. So the function becomes: $f(x) = x^3 - 7x^2 + x^{-2}$

Step 2: Differentiate Each Term

Now, apply the power rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

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

2. Differentiate $-7x^2$: $\frac{d}{dx}(-7x^2) = -7 \cdot 2x^{2-1} = -14x$

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: $f'(x) = 3x^2 - 14x - 2x^{-3}$

Step 4: Rewrite in Standard Form

You can rewrite $-2x^{-3}$ as $-\frac{2}{x^3}$ to present the derivative in a more standard form: $f'(x) = 3x^2 - 14x - \frac{2}{x^3}$

Thus, the derivative of the function $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$ is: $f'(x) = 3x^2 - 14x - \frac{2}{x^3}$

Would you like more details or have any questions?

Here are some related questions:

1. How do you find the second derivative of $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$?
2. What is the derivative of $f(x) = x^3 + x^{-2}$?
3. How do you apply the power rule to negative exponents?
4. What are the critical points of the function $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$?
5. How does the graph of $f(x) = x^3 - 7x^2 + \frac{1}{x^2}$ look compared to its derivative?

Tip: When differentiating functions with negative exponents, remember to decrease the exponent by one and multiply by the original exponent.