Question

Calcular o valor da derivada 3x/(x2-2)

Answer 1

Olá

$\displaystyle \mathsf{y= \frac{3x}{x^2-2} }$

Tem que utilizar a regra do quociente, dada por:

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


Derivando

$\displaystyle \mathsf{y'= \frac{3\cdot (x^2-2)~-~3x\cdot (2x)}{(x^2-2)^2} }\\\\\\\text{Aplicando a distributiva}\\\\\\\mathsf{y'= \frac{3x^2-6-6x^2}{(x^2-2)^2} }\\\\\\\text{Agrupando os termos em comum}\\\\\\\mathsf{y'= \frac{-3x^2-6}{(x^2-2)^2} }\\\\\\\text{Colocando o (-3) em evidencia no denominador}\\\\\\\boxed{\mathsf{y'= \frac{-3(x^2+2)}{(x^2-2)^2} }}$

Answer 2

Boa tarde WemillySamara

f(x) = 3x/(x² - 2) = u(x)/v(x) 

regra de derivação

(u/v)' = (u'v - uv')/v²

u = 3x, u' = 3
v = x² - 2, v' = 2x

(u/v)' = (u'v - uv')/v²
(u/v)' = (3*(x² - 2) - 3x*2x)/(x² - 2)²
(u/v)' = -(3x² + 6x)/(x² - 2)²