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You want to differentiate the given function:

\[
F(z) = \frac{A + Bz + Cz^2}{z^2}
\]

We can rewrite this expression before differentiating, by breaking it into separate terms:

\[
F(z) = \frac{A}{z^2} + \frac{Bz}{z^2} + \frac{Cz^2}{z^2} = \frac{A}{z^2} + \frac{B}{z} + C
\]

Now, let's differentiate each term with respect to \( z \):

1. The derivative of \( \frac{A}{z^2} \) is \( -\frac{2A}{z^3} \).
2. The derivative of \( \frac{B}{z} \) is \( -\frac{B}{z^2} \).
3. The derivative of \( C \) is 0, since \( C \) is a constant.

Thus, the derivative of \( F(z) \) is:

\[
F'(z) = -\frac{2A}{z^3} - \frac{B}{z^2}
\]

Let me know if you need more details or have any questions!

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Here are 5 related questions you could explore:

1. What is the physical interpretation of this derivative in an applied context?
2. How would you handle differentiation if \( A \), \( B \), or \( C \) were functions of \( z \)?
3. What is the second derivative \( F''(z) \)?
4. How does the behavior of \( F'(z) \) change as \( z \to 0 \) or \( z \to \infty \)?
5. Can you apply the quotient rule directly to differentiate \( F(z) \)? How does it compare?

**Tip:** Simplifying complex fractions before differentiation often makes the process easier and reduces errors.

Differentiate the function F(z) = (A + Bz + Cz^2) / z^2

Learn how to differentiate the function F(z) = (A + Bz + Cz^2) / z^2 using the power rule and quotient rule. The detailed step-by-step solution explains the process and provides insights into the calculus principles involved. Suitable for undergraduate calculus students.

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