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