To find the derivative of the function $f(x) = 4x^3 - 2x + \frac{1}{x^2}$, we will differentiate each term separately.

Step 1: Differentiate each term.

1. The derivative of $4x^3$ is: $\frac{d}{dx}(4x^3) = 12x^2$

2. The derivative of $-2x$ is: $\frac{d}{dx}(-2x) = -2$

3. The derivative of $\frac{1}{x^2}$ is a little more complex. First, we rewrite it as $x^{-2}$, and then use the power rule: $\frac{d}{dx}(x^{-2}) = -2x^{-3}$ or equivalently: $\frac{d}{dx}\left(\frac{1}{x^2}\right) = -\frac{2}{x^3}$

Step 2: Combine all the derivatives.

Now, putting everything together, we get: $f'(x) = 12x^2 - 2 - \frac{2}{x^3}$

So, the derivative is: $f'(x) = 12x^2 - 2 - \frac{2}{x^3}$

Would you like to go over the steps in more detail, or have any further questions about this?

Here are some related questions to explore:

1. How do we apply the power rule to functions involving negative exponents?
2. What happens to the derivative of $\frac{1}{x^n}$ as $n$ changes?
3. How can we simplify expressions involving negative exponents after differentiating?
4. How does the derivative of a polynomial function differ from the derivative of rational functions?
5. What is the second derivative of the function $f(x) = 4x^3 - 2x + \frac{1}{x^2}$?

Tip: Remember to apply the power rule carefully, especially when dealing with negative exponents, and simplify your final result if possible.