# The derivative of a complex function.

Question:

Find all points at which the complex valued function $f$ define by $$f(z)=(2+i)z^3-iz^2+4z-(1+7i)$$ has a derivative.

I know that $z^3$,$z^2$, and $z$ are differentiable everywhere in the domain of $f$, but how can I write my answer formally? Please can somebody help?

Note:I want to solve the problem without using Cauchy-Riemann equations.

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So $z\in \mathbb{C}$, I guess? – draks ... Mar 1 '12 at 7:40
@draks: yes $z$ is in $\mathbb C$ – Hassan Muhammad Mar 1 '12 at 9:28
Hint: put z=x+iy and then get f(x+iy)=U(x,y)+iV(x,y). You can check the question and follow ways in answers math.stackexchange.com/questions/102623/… – Mathlover Mar 1 '12 at 9:54
@Mathlover: A good hint, but I don't want to use Cauchy-Riemann equations. – Hassan Muhammad Mar 1 '12 at 10:20
You can just write the definition of a derivative (ie. take $z\in\mathbb{C}$ and find $\lim_{h\rightarrow 0}{f(z+h)-f(z) \over h}$), considering it's a polynomial it shouldn't be too hard to find the answer. – Najib Idrissi Mar 1 '12 at 12:23

yes it make sense, but supposing I do not know how to use Cauchy-Riemann equations in fact I don't want to use C-R equation. If I understand you, since finite linear combination of differentiable functions is also differentiable then the whole function $f$ is differentiable. – Hassan Muhammad Mar 1 '12 at 9:36