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SMK Teknologi dan Rekayasa

Solusi Paket 59 UN 2011 Matematika SMK Teknologi (39)


Jika\ f(x)=\frac{-x+3}{4x-1},\ dengan\ x\ne \frac{1}{4}\ maka\ turunan\ pertama\ dari\ f(x)\ adalah\\f{}'(x)=\ ....\\\\A.\ \ \frac{11}{(4x-1)^{2}}\ \ \ \ \ \ \ \ \ \ B.\ \ \frac{-4}{(4x-1)^{2}}\ \ \ \ \ \ \ \ \ \ C.\ \ \frac{-8x+13}{(4x-1)^{2}}\ \ \ \ \ \ \ \ \ \ D.\ \ \frac{-5}{(4x-1)^{2}}\ \ \ \ \ \ \ \ \ \ E.\ \ \frac{-11}{(4x-1)^{2}}
Jawaban: E
u(x)=-x+3\ \ \ \ \ \to \ \ \ \ \ u{}'(x)=-1+0=-1\\v(x)=4x-1\ \ \ \ \ \ \ \to \ \ \ \ \ v{}'(x)=4-0=4
\begin{array}{rcl}f{}'(x)&=&\frac{u{}'.v-u.v{}'}{v^{2}}\\\\&=&\frac{-1.(4x-1)-(-x+3).4}{(4x-1)^{2}}\\\\&=&\frac{-4x+1+4x-12}{(4x-1)^{2}}\\\\&=&\frac{-11}{(4x-1)^{2}}\end{array}
Trik jitu:
f(x)=\frac{ax+b}{cx+d}\ \ \ \ \ \to \ \ \ \ \ f{}'(x)=\frac{a.d-b.c}{(cx+d)^{2}}
f(x)=\frac{-x+3}{4x-1}\qquad \rightarrow \qquad a=-1,\ b=3,\ c=4,\ d=-1
Karena setiap pilihan jawaban memiliki penyebut yang sama maka cukup ditentukan pembilangnya saja, yaitu:
a.d-b.c=-1.(-1)-4.3=1-12=-11

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