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Derivada de funciones de variable
compleja

To compute the derivative of the complex function \( f(z) = 4z^3 - (3 + i)z^2 - 5z + 4 \), we will apply the standard rules for differentiation with respect to the complex variable \( z \), using the linearity of the derivative and the power rule.

### Function:
\[
f(z) = 4z^3 - (3 + i)z^2 - 5z + 4
\]

### Derivative:
1. The derivative of \( 4z^3 \) is \( 12z^2 \) (using the power rule: \( \frac{d}{dz}z^n = n z^{n-1} \)).
2. The derivative of \( -(3 + i)z^2 \) is \( -2(3 + i)z \).
3. The derivative of \( -5z \) is \( -5 \).
4. The derivative of the constant \( 4 \) is \( 0 \).

Thus, the derivative \( f'(z) \) is:
\[
f'(z) = 12z^2 - 2(3 + i)z - 5
\]

Would you like further details on this process or a different related calculation?

Here are five related questions for further exploration:
1. How would you compute higher-order derivatives of this function?
2. What are the critical points of this function in the complex plane?
3. Can you find the zeros of the derivative \( f'(z) \)?
4. How does the behavior of this function differ for real and imaginary values of \( z \)?
5. How can the derivative be applied to study the local behavior of this complex function?

**Tip:** For complex-valued functions, the differentiation rules are the same as for real functions, but keep in mind that the variables and constants may include imaginary components.

Derivada de funciones de variable compleja

Learn how to compute the derivative of the complex function f(z) = 4z^3 - (3 + i)z^2 - 5z + 4 using the power rule. This step-by-step guide covers key concepts in complex analysis, suitable for university-level students.

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