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