Question

3. Let f(z) = zc where c is a complex number. Assume that the domain of f is the whole complex plane except the negative real numbers. a) What is the derivative of f? b) Let g(z) = cz. Find the derivative of g.

3. Let f(z) = zº where c is a complex number. Assume that the domain of f is the whole complex plane except the negative real numbers. a) What is the derivative of f? b) Let g(z) = c+. Find the derivative of g.

Answer 1

if(2) = zº, CEC we know if'(2) ie derivative of f(2) fl(2) = lim f(2+$2), f(2) AZ AZ 12+ sz) C_ f(2) = lim - oz z >0 C-1 2 Z + = tim zf + c.2°toz telev.2 (0:2) --| 12 AZ O DZ C - CCGU, 2 •D2T (-2 .02 +- -. - . Z cz to lim AZ to - C2CI+otori CZ Eczcy Anes other Method f(2)=zC taking log on both sides log (F(2)) = clog(2) diff.au.roto z tejflZ) = { f(2) fles = f(2) x azeiten (f/2F C Z C Am

f(2)= CZ taring logon both sides log. (f(2)) = log1c?) 10g (F(21) = zlogi) ; CEC differentiate warto z ti f(Z) = Z xo + logic) f'(2) = f(2) loge f'(2) = (² loge Arsenler