Question

calcule f' (x) se
f(x)= x^2 - 1/
x+1

Answer 1

$f(x)=\dfrac{x^2-1}{x+1}$


Forma 1: Usando a regra da derivada do quociente:

$f'(x)=\left(\dfrac{x^2-1}{x+1}\right)'\\\\\\ =\dfrac{(x^2-1)'\cdot (x+1)-(x^2-1)\cdot (x+1)'}{(x+1)^2}\\\\\\ =\dfrac{2x\cdot (x+1)-(x^2-1)\cdot 1}{(x+1)^2}\\\\\\ =\dfrac{2x^2+2x-x^2+1}{(x+1)^2}\\\\\\ =\dfrac{x^2+2x+1}{(x+1)^2}\\\\\\ =\dfrac{(x+1)^2}{(x+1)^2}\\\\\\ =1~~~~\text{para }x\ne 1$


Forma 2: Simplificando a expressão da função antes de derivar:

$f(x)=\dfrac{x^2-1}{x+1}\\\\\\ f(x)=\dfrac{(x+1)(x-1)}{x+1}\\\\\\ f(x)=x-1~~~~~~\text{para }x\ne 1$


Então, a derivada é

$f'(x)=(x-1)'\\\\ \boxed{\begin{array}{c}f'(x)=1 \end{array}}$


Dúvidas? Comente.


Bons estudos! :-)