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