by-diffention-find-f-z-of-f-z-z-1-3-

Question Number 201393 by mokys last updated on 05/Dec/23

b y d i f f e n t i o n f i n d f^{'} (z) o f f (z) = \sqrt[3]{z}

Commented by mokys last updated on 05/Dec/23

h o w c a n s o l v e t h i s

Answered by aleks041103 last updated on 05/Dec/23

t h i s i s t r i v i a l

f (z) = \sqrt[3]{z} = z^{1 / 3}

f^{'} (z) = \underset{h \to 0}{l i m} \frac{f (z + h) - f (z)}{h} =

= \underset{h \to 0}{l i m} \frac{{(z + h)}^{1 / 3} - z^{1 / 3}}{h} =

= \underset{h \to 0}{l i m} \frac{e^{l n (z + h) / 3} - e^{l n (z) / 3}}{h} =

= e^{l n (z) / 3} \underset{h \to 0}{l i m} \frac{e^{l n (1 + h / z) / 3} - 1}{h} =

= z^{1 / 3} \underset{h \to 0}{l i m} \frac{e^{\frac{h / z + o (h / z)}{3}} - 1}{h} =

= z^{1 / 3} \underset{h \to 0}{l i m} \frac{(1 + \frac{h}{3 z} + o (\frac{h}{z}) + o (\frac{h}{3 z} + o (\frac{h}{3 z}))) - 1}{h} =

= z^{1 / 3} \underset{h \to 0}{l i m} (\frac{1}{3 z} + \frac{1}{z} \frac{o (h / z)}{h / z}) =

(i f f z \neq 0) = \frac{z^{1 / 3}}{3 z} + \frac{z^{1 / 3}}{z} \underset{g \to 0}{l i m} \frac{o (g)}{g} =

= \frac{1}{3} z^{- 2 / 3}

\Rightarrow f^{'} (z) = {\begin{cases} \frac{1}{3 \sqrt[3]{z^{2}}}, z \neq 0 \\ u n d e f f ., z = 0 \end{cases}