|
|
|
|
|
|
|
|
|
|
|
|
Turunan implisit dari $$$\ln\left(y\right) = x \ln\left(2\right)$$$ terhadap $$$x$$$
Masukan Anda
Temukan $$$\frac{d}{dx} \left(\ln\left(y\right) = x \ln\left(2\right)\right)$$$.
Solusi
Turunkan secara terpisah kedua ruas persamaan (anggap $$$y$$$ sebagai fungsi dari $$$x$$$): $$$\frac{d}{dx} \left(\ln\left(y{\left(x \right)}\right)\right) = \frac{d}{dx} \left(x \ln\left(2\right)\right)$$$.
Turunkan ruas kiri dari persamaan.
Fungsi $$$\ln\left(y{\left(x \right)}\right)$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \ln\left(u\right)$$$ dan $$$g{\left(x \right)} = y{\left(x \right)}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(y{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(y{\left(x \right)}\right)\right)}$$Turunan dari logaritma natural adalah $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$Kembalikan ke variabel semula:
$$\frac{\frac{d}{dx} \left(y{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(y{\left(x \right)}\right)}{{\color{red}\left(y{\left(x \right)}\right)}}$$Dengan demikian, $$$\frac{d}{dx} \left(\ln\left(y{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(y{\left(x \right)}\right)}{y{\left(x \right)}}$$$.
Turunkan ruas kanan persamaan.
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 $$$c = \ln\left(2\right)$$$ dan $$$f{\left(x \right)} = x$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \ln\left(2\right)\right)\right)} = {\color{red}\left(\ln\left(2\right) \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\ln\left(2\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = \ln\left(2\right) {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(x \ln\left(2\right)\right) = \ln\left(2\right)$$$.
Dengan demikian, kita memperoleh persamaan linier berikut terhadap turunan: $$$\frac{\frac{dy}{dx}}{y} = \ln\left(2\right)$$$.
Dengan menyelesaikannya, kita memperoleh bahwa $$$\frac{dy}{dx} = y \ln\left(2\right)$$$.
Jawaban
$$$\frac{dy}{dx} = y \ln\left(2\right)$$$A