Question

Determine as derivadas das funções abaixo.
Y= X³+2X²+5/2X+3

Answer 1

Regra do quociente:

$\boxed{\boxed{\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{f'(x)g(x)-g'(x)f(x)}{[g(x)]^{2}}}}$
________________________________

$f(x)=x^{3}+2x^{2}+5~~~\therefore~~~~\boxed{\boxed{f'(x)=3x^{2}+4x}}\\\\g(x)=2x+3~~~\therefore~~~\boxed{\boxed{g'(x)=2}}$

Aplicando a regra do quociente pra achar a derivada de y em relação a x:

$y'=\dfrac{f'(x)g(x)-g'(x)f(x)}{(g(x))^{2}}\\\\\\y'=\dfrac{(3x^{2}+4x)\cdot(2x+3)-2\cdot(x^{3}+2x^{2}+5)}{(2x+3)^{2}}\\\\\\y'=\dfrac{6x^{3}+9x^{2}+8x^{2}+12x-2x^{3}-4x^{2}-10}{(2x+3)^{2}}\\\\\\\boxed{\boxed{y'=\dfrac{4x^{3}+13x^{2}+12x-10}{(2x+3)^{2}}}}$