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Turunan dari $$$x^{2 x}$$$
Kalkulator terkait: Kalkulator Turunan
Masukan Anda
Temukan $$$\frac{d}{dx} \left(x^{2 x}\right)$$$.
Solusi
Misalkan $$$H{\left(x \right)} = x^{2 x}$$$.
Ambil logaritma pada kedua ruas: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{2 x}\right)$$$
Tulis ulang ruas kanan menggunakan sifat-sifat logaritma: $$$\ln\left(H{\left(x \right)}\right) = 2 x \ln\left(x\right)$$$.
Diferensiasikan secara terpisah kedua sisi persamaan: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(2 x \ln\left(x\right)\right)$$$.
Turunkan ruas kiri dari persamaan.
Fungsi $$$\ln\left(H{\left(x \right)}\right)$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \ln\left(u\right)$$$ dan $$$g{\left(x \right)} = H{\left(x \right)}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$Turunan dari logaritma natural adalah $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$Kembalikan ke variabel semula:
$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$Dengan demikian, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.
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 $$$c = 2$$$ dan $$$f{\left(x \right)} = x \ln\left(x\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 x \ln\left(x\right)\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(x \ln\left(x\right)\right)\right)}$$Terapkan aturan hasil kali $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ pada $$$f{\left(x \right)} = x$$$ dan $$$g{\left(x \right)} = \ln\left(x\right)$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(x \ln\left(x\right)\right)\right)} = 2 {\color{red}\left(\frac{d}{dx} \left(x\right) \ln\left(x\right) + x \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$Turunan dari logaritma natural adalah $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$2 x {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} + 2 \ln\left(x\right) \frac{d}{dx} \left(x\right) = 2 x {\color{red}\left(\frac{1}{x}\right)} + 2 \ln\left(x\right) \frac{d}{dx} \left(x\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$2 \ln\left(x\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + 2 = 2 \ln\left(x\right) {\color{red}\left(1\right)} + 2$$Dengan demikian, $$$\frac{d}{dx} \left(2 x \ln\left(x\right)\right) = 2 \ln\left(x\right) + 2$$$.
Dengan demikian, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = 2 \ln\left(x\right) + 2$$$.
Oleh karena itu, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(2 \ln\left(x\right) + 2\right) H{\left(x \right)} = 2 x^{2 x} \left(\ln\left(x\right) + 1\right)$$$.
Jawaban
$$$\frac{d}{dx} \left(x^{2 x}\right) = 2 x^{2 x} \left(\ln\left(x\right) + 1\right)$$$A