Turunan dari $$$\ln\left(32 y\right)$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dy} \left(\ln\left(32 y\right)\right)$$$.
Solusi
Fungsi $$$\ln\left(32 y\right)$$$ merupakan komposisi $$$f{\left(g{\left(y \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \ln\left(u\right)$$$ dan $$$g{\left(y \right)} = 32 y$$$.
Terapkan aturan rantai $$$\frac{d}{dy} \left(f{\left(g{\left(y \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dy} \left(g{\left(y \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dy} \left(\ln\left(32 y\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dy} \left(32 y\right)\right)}$$Turunan dari logaritma natural adalah $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dy} \left(32 y\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dy} \left(32 y\right)$$Kembalikan ke variabel semula:
$$\frac{\frac{d}{dy} \left(32 y\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dy} \left(32 y\right)}{{\color{red}\left(32 y\right)}}$$Terapkan aturan kelipatan konstanta $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$ dengan $$$c = 32$$$ dan $$$f{\left(y \right)} = y$$$:
$$\frac{{\color{red}\left(\frac{d}{dy} \left(32 y\right)\right)}}{32 y} = \frac{{\color{red}\left(32 \frac{d}{dy} \left(y\right)\right)}}{32 y}$$Terapkan aturan pangkat $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{dy} \left(y\right)\right)}}{y} = \frac{{\color{red}\left(1\right)}}{y}$$Dengan demikian, $$$\frac{d}{dy} \left(\ln\left(32 y\right)\right) = \frac{1}{y}$$$.
Jawaban
$$$\frac{d}{dy} \left(\ln\left(32 y\right)\right) = \frac{1}{y}$$$A