Question

Qual a derivada de 3x elevado a 4x? [Derivada (3x^4x)]

Answer 1

Calcular a derivada da função composta de exponenciais:

     $f(x)=3x^{4x}\\\\ f(x)=3\cdot x^{4x}$


Como  f  é uma função exponencial, podemos tomar logaritmos de ambos os lados:

     $\ln\big[f(x)\big]=\ln(3\cdot x^{4x})\\\\ \ln\big[f(x)\big]=\ln(3)+\ln(x^{4x})\\\\ \ln\big[f(x)\big]=\ln(3)+4x\ln(x)$


Derive ambos os lados implicitamente em relação a  x:

     $\dfrac{d}{dx}\Big[\ln\big[f(x)\big]\Big]=\dfrac{d}{dx}\big[\ln(3)+4x\ln(x)\big]\\\\\\ \dfrac{d}{dx}\Big[\ln\big[f(x)\big]\Big]=\dfrac{d}{dx}\big[\ln(3)\big]+\dfrac{d}{dx}\big[4x\ln(x)\big]\\\\\\ \dfrac{1}{f(x)}\cdot \dfrac{d}{dx}\big[f(x)\big]=0+\dfrac{d}{dx}(4x)\cdot \ln(x)+4x\cdot \dfrac{d}{dx}\big[\ln(x)\big]\\\\\\ \dfrac{1}{f(x)}\cdot \dfrac{df}{dx}=4\cdot \ln(x)+4\diagup\!\!\!\!\! x\cdot\dfrac{1}{\diagup\!\!\!\!\! x}\\\\\\ \dfrac{1}{f(x)}\cdot \dfrac{df}{dx}=4\ln(x)+4\\\\\\ \dfrac{df}{dx}=f(x)\cdot \big[4\ln(x)+4\big]\\\\\\ \dfrac{df}{dx}=f(x)\cdot 4\big[\ln(x)+1\big]$


Substituindo de volta a lei da função  f(x),  finalmente encontramos

     $\dfrac{df}{dx}=3x^{4x}\cdot 4\big[\ln(x)+1\big]$

     $\dfrac{df}{dx}=12x^{4x}\cdot \big[\ln(x)+1\big]\quad\longleftarrow\quad\textsf{esta }\mathsf{\acute{e}}\textsf{ a resposta.}$


Bons estudos! :-)