10.6.4 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 \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(2 x^{3}-6 x y^{2}-3 x^{2}+3 y^{2}+x+2\right)+ \\ +i\left(6 x^{2} y-2 y^{3}-6 x y+y\right) . \end{array} \]