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