Pembuktian aturan perkalian dan pembagian

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 Aria Turns

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