10.6.3 Analytic functions

Problem: Prove that function \( f(z)=u(x, y)+i v(x, y) \) is differentiable on the entire complex plane and find its derivative \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(3 x^{3}-6 x y^{2}+2 x^{2}-2 y^{2}-3 x+1\right)+ \\ +i\left(6 x^{2} y-2 y^{3}+4 x y-3 y\right) . \end{array} \]