Question

Calcule a derivada 20 PONTOS

y=$\sqrt{ax} + ( a/ \sqrt{ax} )$

Answer 1

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$\large\texttt{Calcular a derivada da fun\c{c}\~ao:}$

$\large\begin{array}{l} \mathtt{y=\sqrt{ax}+\dfrac{a}{\sqrt{ax}}}\\\\ \mathtt{y=(ax)^{1/2}+\dfrac{a}{(ax)^{1/2}}}\\\\ \mathtt{y=(ax)^{1/2}+a\cdot (ax)^{-1/2}\qquad\quad com~a~constante.} \end{array}$


$\large\begin{array}{l} \texttt{Temos uma soma de fun\c{c}\~oes compostas. Logo, utilizamos}\\\texttt{a Regra da Cadeia:}\\\\ \mathtt{\dfrac{dy}{dx}=\dfrac{d}{dx}\big[(ax)^{1/2}+a\cdot (ax)^{-1/2}\big]}\\\\ \mathtt{\dfrac{dy}{dx}=\dfrac{d}{dx}\big[(ax)^{1/2}\big]+a\cdot \dfrac{d}{dx}\big[(ax)^{-1/2}\big]} \end{array}$

$\large\begin{array}{l} \mathtt{\dfrac{dy}{dx}=\left[\dfrac{1}{2}\cdot (ax)^{(1/2)-1}\cdot \dfrac{d}{dx}(ax)\right ]+a\cdot \left[-\,\dfrac{1}{2}\cdot (ax)^{(-1/2)-1}\cdot \dfrac{d}{dx}(ax)\right ]}\\\\ \mathtt{\dfrac{dy}{dx}=\dfrac{1}{2}\cdot (ax)^{-1/2}\cdot a-\dfrac{1}{2}\,a\cdot (ax)^{-3/2}\cdot a}\\\\ \mathtt{\dfrac{dy}{dx}=\dfrac{1}{2}\,a\cdot \dfrac{1}{(ax)^{1/2}}-\dfrac{1}{2}\,a^2\cdot \dfrac{1}{(ax)^{3/2}}} \end{array}$


$\large\begin{array}{l} \therefore~~\boxed{\begin{array}{c} \mathtt{\dfrac{dy}{dx}=\dfrac{a}{2\sqrt{ax}}-\dfrac{a^2}{2\sqrt{(ax)^{3}}}} \end{array}}\qquad\quad\checkmark \end{array}$


$\large\texttt{Bons estudos! :-)}$


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