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If at any particular point the function does not satisfy the Cauchy-Riemann equations, then its complex derivative cannot exist at that point. So we check the Cauchy Riemann equations and see that $$u_x = 3x^{2}+y^{2}~~,~~u_y = 2xy~~,~~v_x = 2xy~~,~~ v_y = x^{2}+3y^{2}$$

Equipping we see that $$3x^{2}+y^{2} = x^{2}+3y^{2} ~\text{and}~ 2xy = -2xy$$

However my problem came when i tried to solve the equation above, i somehow get that the Cauchy Riemann equations are SATISFIED at the points $(0,0)$, does that suggest that function $z^{2}\overline{z}$ actually has a derivative at $z_0 = 0$? I am really confused, please help.

P.S Maybe some kind souls can show their working for me?

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    Except $(0,0)$ ? if you solve the last system of equation ,I think – 2017-01-23
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    It is complex-differentiable at $z = 0$ with derivative $0$. – 2017-01-23
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    So what should i do, the question seems to suggest that there are no derivative for all $z \in \mathbb{C}$ – 2017-01-23
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    Maybe the question was to prove it for all $z \in \mathbb{C}^*$, the punctured complex plane? – 2017-01-23
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    "Maybe some kind souls can show their working for me?" Which "working"? @levap's comment above fully answers your question, doesn't it? – 2017-01-23
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    @Did Well, i may have committed an error in between my steps, since the question seems to state that no value of $z$ satisfies the equation – 2017-01-23
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    @ilovewt: I didn't check your Cauchy-Riemann equations calculations but the complex derivative of $z^2 \overline{z}$ definitely exist at $z = 0$ (as can be seen without much calculation by considering the limit definition). – 2017-01-23
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    Is this a question from a textbook? If the book says "show that the derivative of function so and so does not exist" then that of course is misleading as levap pointed out that the derivative does exist in the Origin. – 2017-01-23
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    @levap thanks a lot! – 2017-01-23
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    @imranfat I took it from an exercise with the exact words! – 2017-01-23
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    That's odd. Something you want to discuss with your professor and classmates... – 2017-01-23

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