Question

Dê a derivada da função ?$y= 3 ^{5x}$

Answer 1

$\large\begin{array}{l} \textsf{Encontrar a derivada da fun\c{c}\~ao}\\\\ \mathsf{y=3^{5x}}\qquad\textsf{(mas }\mathsf{3=e^{\ell n\,3}}\textsf{)}\\\\ \mathsf{y=(e^{\ell n\,3})^{5x}}\\\\ \mathsf{y=e^{(\ell n\,3)\,\cdot\,5x}}\\\\ \mathsf{y=e^{(5\,\ell n\,3)\,x}} \end{array}$


$\large\begin{array}{l} \textsf{Expressando y como uma fun\c{c}\~ao composta:}\\\\ \left\{\! \begin{array}{l} \mathsf{y=e^u}\\ \mathsf{u=(5\,\ell n\,3)\,x} \end{array} \right. \end{array}$


$\large\begin{array}{l} \textsf{Deriva-se usando a Regra da Cadeia:}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}}\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{du}(e^u)\cdot \dfrac{d}{dx}\big[(5\,\ell n\,3)\,x\big]}\\\\ \mathsf{\dfrac{dy}{dx}=e^u\cdot (5\,\ell n\,3)}\\\\ \mathsf{\dfrac{dy}{dx}=y\cdot (5\,\ell n\,3)}\\\\ \boxed{\begin{array}{c}\mathsf{\dfrac{dy}{dx}=3^{5x}\cdot (5\,\ell n\,3)} \end{array}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.} \end{array}$


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$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags: derivada função composta regra da cadeia exponencial cálculo diferencial