Question

5. Using the limit definition of derivative (and perhaps the previous problem) to show that if P is a polynomial of degree n: P(z)= ao a12 anz", then P'(z)a12a2z + па, 2n-1

Answer 1

Definition of Derivative! a complex Sunction f defined in a non-empty open set D is differentiable for complese differentable) at tot d if the limit lim. }(7) - f (%) exist where I is open subset of c. >Z02 -zo ZZO Le P(Z) = 4 + 4z + .....tayzm let, zo €¢ then tim P (2) - P (20) 7-707 - = lim not azt tanz" (as +9Z+ tan 2) 7-20 - lim (ac-a0) + 24 (7-zo) + ---t am (zon- 272 - 17--Zo) = lim afzazo) + a, (22 - 2 ) + ( 23%) +- fam (20 2-20 offz-2017 Qq (Z-Zo) (2+20) +23 (7-3) (Zizz+2) 220 + a4 (z-Zo) (²2² + Zo + z 2 + 2 t fan/Z-Zo)/27 zzo = lim +--+2 ) 20 a + a₂ 12 470) +9, (7*zzo + Z0) +90 (2²+ Ezott tej = lim (2-2)... an (zn-1 + 2 zo + 2 M-12 220 to 12 7-20 slimno ?" tani m-44 x 220 + -- 27") } - lim çatą (z +20) + Q3 (Zt Z Zo + ZE )+ 04 (2²+ Zzotziz+z? at az (zotzo) & a (totz Zot ză lt . tan (et

ie lim Z7zo .P(Z) - P(28) 2 = a + 2a Zo +393 to to thanh z- zo le for any zo €¢ . lim.no exist to ee . lim .PIZ) - P(2) to 7- Zo ie deed derivative of p(z) earst atzo. - for any zot€ so derivate of P(2) exist for all 76¢ Sine Lim P(2) - P.(20) = 4 +292 zotiazotofnanza Z +7 Z-zo so Pézo) = a + 2ay zot -- than any at z=zo frecer for derivative atz 1 ľ(z) =a + 2az & -tha , za-1 Hence derivative of . P(z) = ao fazt qttartanah p'(z) = 24 +2a 77.-- Ananzaal is