Example Find the derivative of the following function at the given point．

$$f(z) = \frac{(z - i)}{(z + i)}, z = i$$

Answer 1.

$$f'(z) = \frac{(z-i)'(z+i)-(z-i)(z+i)'}{(z+i)^2}$$

$$= \frac{2i}{(z+i)^2}$$

Thus，

$$f'(i) = \frac{2i}{(i+i)^2} = \frac{2i}{-4} = -\frac{i}{2}$$