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Turunan kedua dari $$$e^{3 x}$$$
Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik
Masukan Anda
Temukan $$$\frac{d^{2}}{dx^{2}} \left(e^{3 x}\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(e^{3 x}\right)$$$
Fungsi $$$e^{3 x}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = e^{u}$$$ dan $$$g{\left(x \right)} = 3 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{3 x}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(3 x\right)\right)}$$Turunan dari fungsi eksponensial adalah $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(3 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(3 x\right)$$Kembalikan ke variabel semula:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(3 x\right) = e^{{\color{red}\left(3 x\right)}} \frac{d}{dx} \left(3 x\right)$$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 $$$c = 3$$$ dan $$$f{\left(x \right)} = x$$$:
$$e^{3 x} {\color{red}\left(\frac{d}{dx} \left(3 x\right)\right)} = e^{3 x} {\color{red}\left(3 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 e^{3 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 e^{3 x} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(e^{3 x}\right) = 3 e^{3 x}$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(e^{3 x}\right) = \frac{d}{dx} \left(3 e^{3 x}\right)$$$
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 $$$c = 3$$$ dan $$$f{\left(x \right)} = e^{3 x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(3 e^{3 x}\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(e^{3 x}\right)\right)}$$Fungsi $$$e^{3 x}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = e^{u}$$$ dan $$$g{\left(x \right)} = 3 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(e^{3 x}\right)\right)} = 3 {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(3 x\right)\right)}$$Turunan dari fungsi eksponensial adalah $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$3 {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(3 x\right) = 3 {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(3 x\right)$$Kembalikan ke variabel semula:
$$3 e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(3 x\right) = 3 e^{{\color{red}\left(3 x\right)}} \frac{d}{dx} \left(3 x\right)$$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 $$$c = 3$$$ dan $$$f{\left(x \right)} = x$$$:
$$3 e^{3 x} {\color{red}\left(\frac{d}{dx} \left(3 x\right)\right)} = 3 e^{3 x} {\color{red}\left(3 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$9 e^{3 x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 9 e^{3 x} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(3 e^{3 x}\right) = 9 e^{3 x}$$$.
Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(e^{3 x}\right) = 9 e^{3 x}$$$.
Jawaban
$$$\frac{d^{2}}{dx^{2}} \left(e^{3 x}\right) = 9 e^{3 x}$$$A