Derivada de $$$\ln^{3}\left(u\right)$$$
Calculadoras relacionadas: Calculadora de Derivação Logarítmica, Calculadora de Diferenciação Implícita com Passos
Sua entrada
Encontre $$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right)$$$.
Solução
A função $$$\ln^{3}\left(u\right)$$$ é a composição $$$f{\left(g{\left(u \right)} \right)}$$$ de duas funções $$$f{\left(v \right)} = v^{3}$$$ e $$$g{\left(u \right)} = \ln\left(u\right)$$$.
Aplique a regra da cadeia $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln^{3}\left(u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{3}\right) \frac{d}{du} \left(\ln\left(u\right)\right)\right)}$$Aplique a regra da potência $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ com $$$n = 3$$$:
$${\color{red}\left(\frac{d}{dv} \left(v^{3}\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = {\color{red}\left(3 v^{2}\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$Retorne à variável original:
$$3 {\color{red}\left(v\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right) = 3 {\color{red}\left(\ln\left(u\right)\right)}^{2} \frac{d}{du} \left(\ln\left(u\right)\right)$$A derivada do logaritmo natural é $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$$3 \ln^{2}\left(u\right) {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} = 3 \ln^{2}\left(u\right) {\color{red}\left(\frac{1}{u}\right)}$$Logo, $$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$.
Resposta
$$$\frac{d}{du} \left(\ln^{3}\left(u\right)\right) = \frac{3 \ln^{2}\left(u\right)}{u}$$$A