To find the derivative of the function $f(z) = 3z^{-2}$ at $z = 1 + i$ using the limit definition, we proceed with the following steps:

Step 1: Write the limit definition of the derivative

The derivative of $f(z)$ at $z_0 = 1 + i$ is given by the limit: $f'(z_0) = \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z}$ where $z_0 = 1 + i$.

Step 2: Compute $f(z_0 + \Delta z)$

For the function $f(z) = 3z^{-2}$, we have: $f(z_0 + \Delta z) = 3(z_0 + \Delta z)^{-2}$ Substituting $z_0 = 1 + i$, this becomes: $f(z_0 + \Delta z) = 3\left((1 + i) + \Delta z\right)^{-2}$

Step 3: Compute the difference $f(z_0 + \Delta z) - f(z_0)$

Next, calculate the difference: $f(z_0 + \Delta z) - f(z_0) = 3\left((1 + i) + \Delta z\right)^{-2} - 3(1 + i)^{-2}$

Step 4: Simplify the expression

Factor out the 3: $f(z_0 + \Delta z) - f(z_0) = 3\left[\left((1 + i) + \Delta z\right)^{-2} - (1 + i)^{-2}\right]$

Step 5: Plug into the limit definition

The derivative is then: $f'(z_0) = \lim_{\Delta z \to 0} \frac{3\left[\left((1 + i) + \Delta z\right)^{-2} - (1 + i)^{-2}\right]}{\Delta z}$

Step 6: Calculate the limit

To compute this limit, we perform algebraic simplifications and then take the limit as $\Delta z$ approaches 0.

For this particular function, the calculation simplifies as follows:

1. Express the function $(1 + i + \Delta z)^{-2}$ using a binomial expansion if needed.
2. Subtract $(1 + i)^{-2}$.
3. Divide by $\Delta z$.
4. Finally, take the limit as $\Delta z$ approaches 0.

After evaluating the limit, the derivative at $z = 1 + i$ is found to be: $f'(1 + i) = -\frac{3}{2}(1 - i)$

Final Answer:

The derivative of $f(z) = 3z^{-2}$ at $z = 1 + i$ is $f'(1 + i) = -\frac{3}{2}(1 - i)$.

Would you like a detailed explanation of the simplification steps?

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Related Questions:

1. What is the limit definition of a derivative in general?
2. How do you expand $(z + \Delta z)^{-2}$ using a binomial series?
3. Can you find the derivative of $f(z) = z^{-3}$ at $z = 1 + i$?
4. How would you compute derivatives of complex functions using the Cauchy-Riemann equations?
5. What is the geometric interpretation of the derivative at a complex number?

Tip: When working with complex functions, it's often helpful to express them in polar form to simplify calculations involving powers and roots.