Question

calcule a derivada (x²+1/x²)^4

Answer 1

Calcular a derivada em relação a $x$:

$y=\left(x^{2}+\dfrac{1}{x^{2}} \right )^{4}$


Aplicaremos a Regra da Cadeia, fazendo

$y=u^{4}\;\;\text{ e }\;\;u=x^{2}+\dfrac{1}{x^{2}}:$


$\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot \dfrac{du}{dx}\\ \\ \\ \dfrac{dy}{dx}=\dfrac{d}{du}(u^{4})\cdot \dfrac{d}{dx}\left(x^{2}+\dfrac{1}{x^{2}} \right )\\ \\ \\ \dfrac{dy}{dx}=4u^{4-1}\cdot \left[\dfrac{d}{dx}(x^{2})+\dfrac{d}{dx}\left(\dfrac{1}{x^{2}} \right ) \right ]\\ \\ \\ \dfrac{dy}{dx}=4u^{3}\cdot \left[\dfrac{d}{dx}(x^{2})+\dfrac{d}{dx}(x^{-2}) \right ]\\ \\ \\ \dfrac{dy}{dx}=4u^{3}\cdot \left[2x^{2-1}+(-2)x^{-2-1} \right ]\\ \\ \\ \dfrac{dy}{dx}=4u^{3}\cdot [2x-2x^{-3}]\\ \\ \\ \dfrac{dy}{dx}=4u^{3}\cdot \left[2x-\dfrac{2}{x^{3}} \right ]\\ \\ \\ \boxed{\dfrac{dy}{dx}=4\left(x^{2}+\dfrac{1}{x^{2}} \right )^{3}\cdot \left(2x-\dfrac{2}{x^{3}} \right )}$