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Turunan dari $$$2 e^{2 x}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dx} \left(2 e^{2 x}\right)$$$.
Solusi
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 $$$f{\left(x \right)} = e^{2 x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 e^{2 x}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(e^{2 x}\right)\right)}$$Fungsi $$$e^{2 x}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = e^{u}$$$ dan $$$g{\left(x \right)} = 2 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(e^{2 x}\right)\right)} = 2 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(2 x\right)\right)}$$Turunan dari fungsi eksponensial adalah $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$2 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(2 x\right) = 2 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(2 x\right)$$Kembalikan ke variabel semula:
$$2 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(2 x\right) = 2 e^{{\color{red}\left(2 x\right)}} \frac{d}{dx} \left(2 x\right)$$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 $$$f{\left(x \right)} = x$$$:
$$2 e^{2 x} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = 2 e^{2 x} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$4 e^{2 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 4 e^{2 x} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(2 e^{2 x}\right) = 4 e^{2 x}$$$.
Jawaban
$$$\frac{d}{dx} \left(2 e^{2 x}\right) = 4 e^{2 x}$$$A