Question

Use a proof by induction to show that

Answer 1

Solution dh We have to prove Page No 1 L { t f(x)} = c d fes) = (High difcs) dsn where L{f(x)} = f(s) We prove it by induction (Induction applying on n') Step. I for n=1 fcs) = 1 {fet)} &cs) = sest sets dt differentiate both sides with respect to s & f(s) = d hest fets ht - ds & Now by using Leibniz rule of differentiation under integral sign, we get als fes? = ja cest f(xs). dit & fcs) = je (t) est fets at dat efficas do .. (v) endt sostet fet33 dt = (-1) de

i. Result is true for nal Page No. 2) lie. Let + (x)} = (-133 d' = (-1) df Step II - Assuming that the theorem is true for nak then l {t fet)} = (t el post t fets dt = (-18k dit ® dk Step I - We have to prove that the statement is true for n=kt) now consider from D se stu' feto oltz Gatge de differentiate both side with respect to s. we get a fost efets dt = a cooke dit dsk again by using Leibniz rule of different- under integral sign, we get -lation KH cuole , fosso con est uke for de - 8-t) est tf(A) de ich do post-15%85+ 4K9464) dt

LAULU pageNo.3 teftat • (-i(H) dr d skit (-17 (e da coger opp casa est secondet fet *cs) de KH KH dskH 1444 Kt d f (A) Test KH. e [t fet) dt = d sky Le the fiets} = Co jest jak - a result is true for nakty all three steps of Induction are completed 2. Khe conclude that for any ne N {th f(t)} = (ha