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The function given is \( r(z) = z^{-3} - z^{1/2} \), and we need to differentiate it with respect to \( z \).

### Step-by-step differentiation:
1. The derivative of \( z^{-3} \) using the power rule \( \frac{d}{dz} z^n = n z^{n-1} \) is:
   \[
   \frac{d}{dz} z^{-3} = -3 z^{-4}
   \]

2. The derivative of \( z^{1/2} \) (or \( z^{0.5} \)) is:
   \[
   \frac{d}{dz} z^{1/2} = \frac{1}{2} z^{-1/2}
   \]

### Therefore, the derivative \( r'(z) \) is:
\[
r'(z) = -3 z^{-4} - \frac{1}{2} z^{-1/2}
\]

Would you like more details, or do you have any further questions?

Here are some related questions for further practice:
1. What is the derivative of \( z^n \) for any real number \( n \)?
2. How would you find the second derivative of \( r(z) \)?
3. What is the derivative of \( z^{-2} + z^3 \)?
4. Can you differentiate the function \( r(z) = z^{-1} + z^{1/3} \)?
5. How does the chain rule apply to differentiating more complex functions involving powers of \( z \)?

**Tip:** Always remember that the power rule applies for any real or fractional exponent, which simplifies differentiation.

Differentiate the function r(z) = z^(-3) - z^(1/2).

Learn how to differentiate the function r(z) = z^(-3) - z^(1/2) using the power rule of calculus. The solution breaks down step-by-step, ideal for high school students studying calculus in Grades 10-12.

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