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Turunan dari $$$\ln^{3}\left(x\right)$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right)$$$.
Solusi
Fungsi $$$\ln^{3}\left(x\right)$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = u^{3}$$$ dan $$$g{\left(x \right)} = \ln\left(x\right)$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln^{3}\left(x\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ dengan $$$n = 3$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$Kembalikan ke variabel semula:
$$3 {\color{red}\left(u\right)}^{2} \frac{d}{dx} \left(\ln\left(x\right)\right) = 3 {\color{red}\left(\ln\left(x\right)\right)}^{2} \frac{d}{dx} \left(\ln\left(x\right)\right)$$Turunan dari logaritma natural adalah $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$3 \ln^{2}\left(x\right) {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = 3 \ln^{2}\left(x\right) {\color{red}\left(\frac{1}{x}\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right) = \frac{3 \ln^{2}\left(x\right)}{x}$$$.
Jawaban
$$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right) = \frac{3 \ln^{2}\left(x\right)}{x}$$$A