To find the derivative $f'(z)$ of the complex-valued function $f(z) = u(x, y) + i v(x, y)$, we use the Cauchy-Riemann equations and the formula:

$f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x},$ provided the Cauchy-Riemann equations hold. Let us compute step-by-step.

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Step 1: Split $f(z)$ into real and imaginary parts

Given: $f(z) = \big( 2x - x^3 + 3xy^2 \big) + i \big( 2y - 3x^2y + y^3 + c \big),$ where: $u(x, y) = 2x - x^3 + 3xy^2, \quad v(x, y) = 2y - 3x^2y + y^3 + c.$

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Step 2: Compute partial derivatives

Derivatives of $u(x, y)$:

$\frac{\partial u}{\partial x} = 2 - 3x^2 + 3y^2, \quad \frac{\partial u}{\partial y} = 6xy.$

Derivatives of $v(x, y)$:

$\frac{\partial v}{\partial x} = -6xy, \quad \frac{\partial v}{\partial y} = 2 - 3x^2 + 3y^2.$

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Step 3: Check the Cauchy-Riemann equations

The Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.$

1. For the first equation: $\frac{\partial u}{\partial x} = 2 - 3x^2 + 3y^2, \quad \frac{\partial v}{\partial y} = 2 - 3x^2 + 3y^2.$ This holds.

2. For the second equation: $\frac{\partial u}{\partial y} = 6xy, \quad -\frac{\partial v}{\partial x} = -(-6xy) = 6xy.$ This also holds.

Thus, the Cauchy-Riemann equations are satisfied.

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Step 4: Compute $f'(z)$

Using: $f'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x},$ we substitute: $\frac{\partial u}{\partial x} = 2 - 3x^2 + 3y^2, \quad \frac{\partial v}{\partial x} = -6xy.$

Thus: $f'(z) = \big( 2 - 3x^2 + 3y^2 \big) + i \big( -6xy \big).$

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Final Answer:

$f'(z) = (2 - 3x^2 + 3y^2) - 6xy i.$

Let me know if you'd like further clarifications or additional examples! 😊

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Related Questions:

1. What are the Cauchy-Riemann equations, and how do they ensure analyticity?
2. How do we interpret $f'(z)$ geometrically in complex analysis?
3. What is the significance of a function satisfying the Cauchy-Riemann equations?
4. How do partial derivatives relate to the real and imaginary components of $f(z)$?
5. What happens if $c$ in the given function is not constant?

Tip: Always verify the Cauchy-Riemann equations before assuming a function is differentiable in the complex sense!