How to differentiate $f(x)=z^i$

Question

I have a question to solve which goes like this: Find the derivative of the principal value $z^i$ at the point $z=(1 + i)$.

So far I have put $z$ into exponential form so $z=re^{i\theta}$, then after taking it to i'th power and differentiating it I got $f'(x)=i \frac{z^i}{|z|} - z^i$ but now Im stuck

Perhaps I went in a wrong direction initially

Answer 1

The (branches of the) complex logarithm are defined by $(\log z)'=1/z$. Since you are only interested in a single point, we can safely differentiate:

$$f(z)=z^i=\exp(i\log z)\quad\implies\quad f'(z)=i\exp(i\log z)\cdot\frac1z=iz^{i-1}.$$