Dalam kalkulus , pendiferensialan logaritmik adalah metode untuk mencari turunan suatu fungsi dengan menggunakan turunan logaritmik dari fungsi
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{\displaystyle (\ln f(x))'={\frac {f'(x)}{f(x)}}\quad \implies \quad f'(x)=f(x)\cdot (\ln f(x))'}
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{\displaystyle f'(x)=\left(e^{\ln f(x)}\right)'=e^{\ln f(x)}\cdot (\ln f(x))'}
Teknik ini biasanya digunakan pada kasus dimana lebih mudah untuk mencari turunan logaritmik dari suatu fungsi, dibandingkan menurunkan fungsi tersebut secara langsung. Hal ini umumnya terjadi pada kasus dimana fungsinya terdiri dari perkalian beberapa suku, sehingga transformasi logaritmik akan mengubah operasi perkalian tersebut menjadi penjumlahan (yang tentunya lebih mudah untuk dicari turunannya). Teknik ini juga berguna ketika diterapkan pada fungsi yang dipangkatkan dengan suatu fungsi lain. Metode pendiferensialan logaritmik bergantung kepada kaidah rantai beserta sifat-sifat dari logaritma (khususnya logaritma alami , yaitu logaritma dengan basis e [2] [3]
Metode ini digunakan sebab sifat-sifat dari logaritma memberikan jalan cepat untuk menyederhanakan fungsi-fungsi rumit yang akan dicari turunannya.[4] [3]
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{\displaystyle \ln(ab)=\ln(a)+\ln(b),\qquad \ln \left({\frac {a}{b}}\right)=\ln(a)-\ln(b),\qquad \ln(a^{b})=b\ln(a).}
Untuk mencari turunan dari hasil kali dua fungsi
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{\displaystyle y(x)=f(x)\cdot g(x)}
logaritma alami , sehingga operasi perkaliannya berubah menjadi penjumlahan
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{\displaystyle {\begin{aligned}\ln \left(y(x)\right)&=\ln \left(f(x)\cdot g(x)\right)\\&=\ln \left(f(x)\right)+\ln \left(g(x)\right)\end{aligned}}}
kaidah rantai dan kaidah penjumlahan , maka diperoleh[5]
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{\displaystyle {\begin{aligned}{\frac {y'(x)}{y(x)}}&={\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\\y'(x)&=y(x)\cdot \left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f(x)\cdot g(x)\cdot \left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f'(x)\cdot g(x)+f(x)\cdot g'(x)\end{aligned}}}
kaidah darab dalam turunan.
Untuk mencari turunan dari hasil bagi dua fungsi
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{\displaystyle y(x)={\dfrac {f(x)}{g(x)}}}
logaritma alami , sehingga operasi pembagiannya berubah menjadi pengurangan
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{\displaystyle {\begin{aligned}\ln \left(y(x)\right)&=\ln \left({\dfrac {f(x)}{g(x)}}\right)\\&=\ln \left(f(x)\right)-\ln \left(g(x)\right)\end{aligned}}}
kaidah rantai dan kaidah penjumlahan , maka diperoleh
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{\displaystyle {\begin{aligned}{\frac {y'(x)}{y(x)}}&={\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}\\y'(x)&=y(x)\cdot \left({\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}\right)\\&={\dfrac {f(x)}{g(x)}}\cdot \left({\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}\right)\\&={\frac {f'(x)}{g(x)}}-{\dfrac {f(x)}{g(x)}}\cdot {\frac {g'(x)}{g(x)}}\\&={\frac {f'(x)\cdot g(x)}{\left(g(x)\right)^{2}}}-{\frac {f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}\end{aligned}}}
kaidah hasil-bagi dalam turunan.
Eksponen berupa fungsi
sunting
Apabila fungsinya memiliki bentuk umum
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{\displaystyle y(x)=f(x)^{g(x)}}
logaritma alami pada kedua ruas, operasi exponen yang ada akan menjadi perkalian, sehingga didapatkan
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{\displaystyle {\begin{aligned}\ln \left(y(x)\right)&=\ln \left(f(x)^{g(x)}\right)\\&=g(x)\cdot \ln \left(f(x)\right)\end{aligned}}}
kaidah rantai dan kaidah darab , maka diperoleh
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{\displaystyle {\begin{aligned}{\frac {y'(x)}{y(x)}}&=g'(x)\cdot \ln \left(f(x)\right)+g(x)\cdot {\frac {f'(x)}{f(x)}}\\y'(x)&=y(x)\cdot \left(g'(x)\cdot \ln \left(f(x)\right)+g(x)\cdot {\frac {f'(x)}{f(x)}}\right)\\&=f(x)^{g(x)}\cdot \left(g'(x)\cdot \ln \left(f(x)\right)+g(x)\cdot {\frac {f'(x)}{f(x)}}\right)\\\end{aligned}}}
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{\displaystyle y(x)=e^{g(x)\cdot \ln \left(f(x)\right)}}
kaidah rantai .
Perumuman eksponen fungsi
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Dengan menggunakan notasi Pi kapital , misalkan
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{\displaystyle {\begin{aligned}y(x)&=(f_{1}(x))^{g_{1}(x)}\cdot (f_{2}(x))^{g_{2}(x)}\cdot \ldots \cdot (f_{n}(x))^{g_{n}(x)}\\y(x)&=\prod _{i\,=\,1}^{n}f_{i}(x)^{g_{i}(x)}\end{aligned}}}
Dengan mengenakan logaritma alami pada kedua ruas, operasi perkaliannya akan berubah menjadi operasi penjumlahan, sehingga notasi Pi kapital di atas akan berubah menjadi notasi sigma kapital .
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{\displaystyle {\begin{aligned}y(x)&=f_{1}(x)^{g_{1}(x)}\cdot f_{2}(x)^{g_{2}(x)}\cdot \ldots \cdot f_{n}(x)^{g_{n}(x)}\\\ln \left(y(x)\right)&=\ln \left((f_{1}(x))^{g_{1}(x)}\cdot (f_{2}(x))^{g_{2}(x)}\cdot \ldots \cdot (f_{n}(x))^{g_{n}(x)}\right)\\&=\ln \left(f_{1}(x)^{g_{1}(x)}\right)+\ln \left(f_{2}(x)^{g_{2}(x)}\right)+\ldots +\ln \left(f_{n}(x)^{g_{n}(x)}\right)\\&=g_{1}(x)\cdot \ln \left(f_{1}(x)\right)+g_{2}(x)\cdot \ln \left(f_{2}(x)\right)+\ldots +g_{n}(x)\cdot \ln \left(f_{n}(x)\right)\\&=\sum _{i\,=\,1}^{n}g_{i}(x)\cdot \ln \left(f_{i}(x)\right)\end{aligned}}}
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{\displaystyle {\begin{aligned}{\frac {y'(x)}{y(x)}}&=\sum _{i\,=\,1}^{n}\left(g_{i}'(x)\cdot \ln \left(f_{i}(x)\right)+g_{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)}}\right)\\y'(x)&=y(x)\cdot \sum _{i\,=\,1}^{n}\left(g_{i}'(x)\cdot \ln \left(f_{i}(x)\right)+g_{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)}}\right)\\&=\left(\prod _{i\,=\,1}^{n}f_{i}(x)^{g_{i}(x)}\right)\cdot \sum _{i\,=\,1}^{n}\left(g_{i}'(x)\cdot \ln \left(f_{i}(x)\right)+g_{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)}}\right)\end{aligned}}}
