Question

valer nota da prova derivada de F(x) 2X-1/3x+3

Answer 1

$\displaystyle f(x)=\frac{2x-1}{3x+3}$

Regra do quociente:
Considerando que u e v são funções deriváveis:
$\displaystyle \frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-v'u}{v^2}$
logo:
$\displaystyle \frac{d}{dx}\left(\frac{2x-1}{3x+3}\right)=\frac{(2x-1)'(3x+3)-(3x+3)'(2x-1)}{(3x+3)^2}\\\\i)~~(2x-1)'=\frac{d}{dx}(2x-1)=2\\\\ii)~(3x+3)'=\frac{d}{dx}(3x+3)=3\\\\\frac{d}{dx}\left(\frac{2x-1}{3x+3}\right)=\frac{2(3x+3)-3(2x-1)}{(3x+3)^2}=\frac{6x+6-6x+3}{(3x+3)^2}=\frac{9}{(3x+3)^2}\\\\=\frac{9}{9x^2+18x+9}=\frac{9}{9}\cdot\frac{1}{x^2+2x+1}=\boxed{\frac{1}{x^2+2x+1}}$