Segunda derivada de $$$2^{n}$$$
Calculadoras relacionadas: Calculadora de Derivadas, Calculadora de Derivação Logarítmica
Sua entrada
Encontre $$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right)$$$.
Solução
Encontre a primeira derivada $$$\frac{d}{dn} \left(2^{n}\right)$$$
Aplique a regra exponencial $$$\frac{d}{dn} \left(m^{n}\right) = m^{n} \ln\left(m\right)$$$ com $$$m = 2$$$:
$${\color{red}\left(\frac{d}{dn} \left(2^{n}\right)\right)} = {\color{red}\left(2^{n} \ln\left(2\right)\right)}$$Logo, $$$\frac{d}{dn} \left(2^{n}\right) = 2^{n} \ln\left(2\right)$$$.
Em seguida, $$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = \frac{d}{dn} \left(2^{n} \ln\left(2\right)\right)$$$
Aplique a regra da constante multiplicativa $$$\frac{d}{dn} \left(c f{\left(n \right)}\right) = c \frac{d}{dn} \left(f{\left(n \right)}\right)$$$ com $$$c = \ln\left(2\right)$$$ e $$$f{\left(n \right)} = 2^{n}$$$:
$${\color{red}\left(\frac{d}{dn} \left(2^{n} \ln\left(2\right)\right)\right)} = {\color{red}\left(\ln\left(2\right) \frac{d}{dn} \left(2^{n}\right)\right)}$$Aplique a regra exponencial $$$\frac{d}{dn} \left(m^{n}\right) = m^{n} \ln\left(m\right)$$$ com $$$m = 2$$$:
$$\ln\left(2\right) {\color{red}\left(\frac{d}{dn} \left(2^{n}\right)\right)} = \ln\left(2\right) {\color{red}\left(2^{n} \ln\left(2\right)\right)}$$Logo, $$$\frac{d}{dn} \left(2^{n} \ln\left(2\right)\right) = 2^{n} \ln^{2}\left(2\right)$$$.
Portanto, $$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = 2^{n} \ln^{2}\left(2\right)$$$.
Resposta
$$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = 2^{n} \ln^{2}\left(2\right)$$$A