Differentiate the function.
F(z) = 
A + Bz + Cz2
z2


We are tasked with differentiating the function \( F(z) = \frac{A + Bz + Cz^2}{z^2} \) with respect to \( z \).

Let's break this down.

### Step 1: Simplify the function
Rewrite the function \( F(z) \) as:

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

This simplifies to:

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

### Step 2: Differentiate term by term
Now differentiate each term with respect to \( z \).

1. For \( \frac{A}{z^2} \):
   \[
   \frac{d}{dz} \left( \frac{A}{z^2} \right) = A \cdot \frac{d}{dz} \left( z^{-2} \right) = -2A z^{-3} = -\frac{2A}{z^3}
   \]

2. For \( \frac{B}{z} \):
   \[
   \frac{d}{dz} \left( \frac{B}{z} \right) = B \cdot \frac{d}{dz} \left( z^{-1} \right) = -B z^{-2} = -\frac{B}{z^2}
   \]

3. For \( C \):
   \[
   \frac{d}{dz}(C) = 0
   \]

### Step 3: Combine the results
Now, combine the differentiated terms:

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

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

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

---

Would you like more details or have any questions on this? Here are five related questions to explore further:

1. How does the chain rule apply to differentiate more complex functions involving powers of \( z \)?
2. What is the significance of simplifying functions before differentiation?
3. How would the derivative change if the powers of \( z \) in the original function were different?
4. How does the differentiation process handle constants like \( A \), \( B \), and \( C \)?
5. Can we apply this method to functions with higher powers or more complex denominators?

**Tip:** Always simplify the function before differentiating to avoid unnecessary complexity!

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

This page offers a step-by-step solution for differentiating the rational function F(z) = (A + Bz + Cz^2) / z^2. Learn key concepts such as the power rule and the linear combination rule, perfect for students in Grades 11-12. Understand how to simplify terms and apply differentiation techniques.

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