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Turunan kedua dari $$$a^{x}$$$ terhadap $$$x$$$
Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik
Masukan Anda
Temukan $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(a^{x}\right)$$$
Terapkan aturan eksponen $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ dengan $$$n = a$$$:
$${\color{red}\left(\frac{d}{dx} \left(a^{x}\right)\right)} = {\color{red}\left(a^{x} \ln\left(a\right)\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(a^{x}\right) = a^{x} \ln\left(a\right)$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = \frac{d}{dx} \left(a^{x} \ln\left(a\right)\right)$$$
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 $$$c = \ln\left(a\right)$$$ dan $$$f{\left(x \right)} = a^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(a^{x} \ln\left(a\right)\right)\right)} = {\color{red}\left(\ln\left(a\right) \frac{d}{dx} \left(a^{x}\right)\right)}$$Terapkan aturan eksponen $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ dengan $$$n = a$$$:
$$\ln\left(a\right) {\color{red}\left(\frac{d}{dx} \left(a^{x}\right)\right)} = \ln\left(a\right) {\color{red}\left(a^{x} \ln\left(a\right)\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(a^{x} \ln\left(a\right)\right) = a^{x} \ln^{2}\left(a\right)$$$.
Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = a^{x} \ln^{2}\left(a\right)$$$.
Jawaban
$$$\frac{d^{2}}{dx^{2}} \left(a^{x}\right) = a^{x} \ln^{2}\left(a\right)$$$A