Page 269 - Buku Paket Kelas 11 Matematika

c. Turunan fungsi f(x) = [u(x)]n dengan u'(x) ada, n bilangan asli.
f'(x)=lim f(x+Dx)-f(x) Dx→0 Dx
= lim [u(x+Dx)]n -[u(x)]n Dx→0 Dx
= lim [u(x+Dx)-u(x)+u(x)]n -[u(x)]n Dx→0
Misal P = [u(x + Dx) – u(x)] = lim [P + u(x)]n -[u(x)]n
(Gunakan Binomial Newton) = lim Pn +CnPn-1[u(x)]+CnPn-2[u(x)]2 +...+Cn P[u(x)]n-1 +[u(x)]n -[u(x)]n
Dx→0 Dx
1 2 n-1
Dx
2 n-2 n-1
Dx→0
= lim Pn +nPn-1[u(x)]+CnPn-2[u(x)]2 +...+Cn P2[u(x)]n-2 +Cn P[u(x)]n-1
Dx→0
= lim P(Pn-1 +nPn-2[u(x)]2 +...+Cn P[u(x)]n-2 +Cn [u(x)]n-1)
Dx
n-2 n-1
Dx
Dx→0 Dx Dx→0 n-2 n-1
Dx→0
= lim P lim(Pn-1 +nPn-2[u(x)]2 +...+Cn P[u(x)]n-2 +Cn [u(x)]n-1)
Karena lim P = lim u(x+Dx)-u(x) = u'(x)
Dx→0 Dx Dx→0
lim P = lim u(x + Dx) – u(x) = 0
= nu'(x)[u(x)]n – 1.
Dx→0 Dx→0 = u'(x)[0 + n[u(x)]]n – 1
Dx
MATEMATIKA 261