Question

obtenha a derivada de cada função a seguir f(x)= In(x)/vx

Answer 1

Calcular a derivada da função

     $y=\dfrac{\ln x}{\sqrt{x}}$


Aqui você deve usar a Regra do Quociente:

     $\dfrac{dy}{dx}=\dfrac{d}{dx}\!\left(\dfrac{\ln x}{\sqrt{x}}\right)\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{d}{dx}(\ln x)\cdot \sqrt{x}-\ln x\cdot \frac{d}{dx}(\sqrt{x})}{(\sqrt{x})^2}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{d}{dx}(\ln x)\cdot \sqrt{x}-\ln x\cdot \frac{d}{dx}(x^{1/2})}{x}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot \sqrt{x}-\ln x\cdot \frac{1}{2}\,x^{(1/2)-1}}{x}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot \sqrt{x}-\ln x\cdot \frac{1}{2}\,x^{-1/2}}{x}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot \sqrt{x}-\ln x\cdot \frac{1}{2x^{1/2}}}{x}$

     $\dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot \sqrt{x}-\ln x\cdot \frac{1}{2\sqrt{x}}}{x}$


Multiplique em cima e embaixo por  $\sqrt{x}:$

     $\dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot \sqrt{x}-\ln x\cdot \frac{1}{2\sqrt{x}}}{x}\cdot \dfrac{\sqrt{x}}{\sqrt{x}}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot (\sqrt{x})^2-\ln x\cdot \frac{1\,\cdot\,\sqrt{x}}{2\sqrt{x}}}{x\sqrt{x}}\\\\\\ \dfrac{dy}{dx}=\dfrac{\frac{1}{x}\cdot x-\ln x\cdot \frac{1}{2}}{x\sqrt{x}}\\\\\\ \dfrac{dy}{dx}=\dfrac{1-\frac{1}{2}\ln x}{x\sqrt{x}}$


Multiplique em cima e embaixo por  $2:$

     $\dfrac{dy}{dx}=\dfrac{(1-\frac{1}{2}\ln x)\cdot 2}{x\sqrt{x}\cdot 2}$

     $\dfrac{dy}{dx}=\dfrac{2-\ln x}{2x\sqrt{x}}\quad\longleftarrow\quad\textsf{esta \'e a resposta.}$


Bons estudos! :-)