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Turunan dari $$$\ln\left(2 u\right)$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{du} \left(\ln\left(2 u\right)\right)$$$.
Solusi
Fungsi $$$\ln\left(2 u\right)$$$ merupakan komposisi $$$f{\left(g{\left(u \right)} \right)}$$$ dari dua fungsi $$$f{\left(v \right)} = \ln\left(v\right)$$$ dan $$$g{\left(u \right)} = 2 u$$$.
Terapkan aturan rantai $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(2 u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(\ln\left(v\right)\right) \frac{d}{du} \left(2 u\right)\right)}$$Turunan dari logaritma natural adalah $$$\frac{d}{dv} \left(\ln\left(v\right)\right) = \frac{1}{v}$$$:
$${\color{red}\left(\frac{d}{dv} \left(\ln\left(v\right)\right)\right)} \frac{d}{du} \left(2 u\right) = {\color{red}\left(\frac{1}{v}\right)} \frac{d}{du} \left(2 u\right)$$Kembalikan ke variabel semula:
$$\frac{\frac{d}{du} \left(2 u\right)}{{\color{red}\left(v\right)}} = \frac{\frac{d}{du} \left(2 u\right)}{{\color{red}\left(2 u\right)}}$$Terapkan aturan kelipatan konstanta $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$ dengan $$$c = 2$$$ dan $$$f{\left(u \right)} = u$$$:
$$\frac{{\color{red}\left(\frac{d}{du} \left(2 u\right)\right)}}{2 u} = \frac{{\color{red}\left(2 \frac{d}{du} \left(u\right)\right)}}{2 u}$$Terapkan aturan pangkat $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{du} \left(u\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{du} \left(u\right)\right)}}{u} = \frac{{\color{red}\left(1\right)}}{u}$$Dengan demikian, $$$\frac{d}{du} \left(\ln\left(2 u\right)\right) = \frac{1}{u}$$$.
Jawaban
$$$\frac{d}{du} \left(\ln\left(2 u\right)\right) = \frac{1}{u}$$$A