Question

use a regra do quociente e calcule a derivada de y=x^2+x-2/x^3+6

Answer 1

Olá



Regra do quociente:

$\displaystyle\mathsf{\left( \frac{f}{g} \right)'~=~ \frac{f'\cdot g~-~f\cdot g'}{g^2} }$





$\displaystyle\mathsf{y~=~ \frac{x^2+x-2}{x^3+6} }\\\\\\\mathsf{y'~=~ \frac{(2x+1)(x^3+6)~-~(x^2+x-2)(3x^2)}{(x^3+6)^2} }\\\\\\\\\text{Simplifica, Aplicando-se a distributiva}\\\\\\\\\mathsf{y'~=~ \frac{2x^4+12x+x^3+6~-~3x^4-3x^3+6x^2}{(x^3+6)^2} }\\\\\\\\\boxed{\mathsf{y'~=~ \frac{-x^4-2x^3+6x^2+12x+6}{(x^3+6)^2} }}$