Pembuktian aturan perkalian dan pembagian

Posted on May 1, 2016 by Nursatria

Blog ini pernah membahas pembuktian aturan rantai, sekarang kita akan membahas pembuktian aturan perkalian dan pembagian pada turunan. Ada banyak cara untuk membuktikan 2 hal tersebut tetapi kita akan menggunakan cara yang paling sederhana, yaitu dengan menggunakan logaritma natural.

Pertama-tama kita buktikan dulu aturan perkalian

Aturan perkalian: Jika f\left(x\right)=g\left(x\right)h\left(x\right) maka berlaku

f'\left(x\right)=g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)

Bukti:

Kita berikan logaritma natural ke f\left(x\right)=g\left(x\right)h\left(x\right) pada kedua sisinya, diperoleh:

\ln f\left(x\right)=\ln\left(g\left(x\right)h\left(x\right)\right)

Berdasarkan sifat logaritma,diperoleh

\ln f\left(x\right)=\ln g\left(x\right)+\ln h\left(x\right)

Selajutnya kita turunkan dengan mengunakan turunan logaritma natural dan aturan rantai, didapat

{\displaystyle \frac{f'\left(x\right)}{f\left(x\right)}=\frac{g'\left(x\right)}{g\left(x\right)}+\frac{h'\left(x\right)}{h\left(x\right)}}

{\displaystyle \frac{f'\left(x\right)}{f\left(x\right)}=\frac{g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}

{\displaystyle f'\left(x\right)=f\left(x\right)\cdot{\displaystyle \frac{g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}}

Ingat f\left(x\right)=g\left(x\right)h\left(x\right)

{\displaystyle f'\left(x\right)=g\left(x\right)h\left(x\right)\cdot{\displaystyle \frac{g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}}

f'\left(x\right)=g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)

\boxempty

Selajutnya aturan pembagian, caranya sama saja

Aturan pembagaian: Jika {\displaystyle f\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}} maka berlaku

{\displaystyle f'\left(x\right)=\frac{g'\left(x\right)h\left(x\right)-h'\left(x\right)g\left(x\right)}{h^{2}\left(x\right)}}

Bukti:

{\displaystyle \ln f\left(x\right)=\ln\frac{g\left(x\right)}{h\left(x\right)}}

\ln f\left(x\right)=\ln g\left(x\right)-\ln h\left(x\right)

{\displaystyle \frac{f'\left(x\right)}{f\left(x\right)}=\frac{g'\left(x\right)}{g\left(x\right)}-\frac{h'\left(x\right)}{h\left(x\right)}}

{\displaystyle \frac{f'\left(x\right)}{f\left(x\right)}=\frac{g'\left(x\right)h\left(x\right)-h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}

{\displaystyle f'\left(x\right)=f\left(x\right)\cdot{\displaystyle \frac{g'\left(x\right)h\left(x\right)-h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}}

{\displaystyle f'\left(x\right)=\frac{g\left(x\right)}{h\left(x\right)}\cdot{\displaystyle \frac{g'\left(x\right)h\left(x\right)+h'\left(x\right)g\left(x\right)}{g\left(x\right)h\left(x\right)}}}

{\displaystyle f'\left(x\right)=\frac{g'\left(x\right)h\left(x\right)-h'\left(x\right)g\left(x\right)}{h^{2}\left(x\right)}}

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About Nursatria

Seorang Alumnus Matematika UGM, dengan ilmu yang didapat ketika kuliah (Padahal sering bolos kuliah :p ), saya menyebarkan virus matematika
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