Question

f(x + h) − f(x))/h, onde f(x) = x^3 + 2x e h é diferente de 0. e f(x)=1/x e h diferente de 0.

Answer 1

1)

$\frac{\left[(x+h)^3+2.(x+h)\right]-\left[x^3+2x\right]}{h}\\\\\\\frac{\left[(x^3+3x^2h+3xh^2+h^3)+(2x+2h)\right]-\left[x^3+2x\right]}{h}\\\\\\\frac{(x^3+3x^2h+3xh^2+h^3)+(2x+2h)-x^3-2x}{h}\\\\\\\frac{3x^2h+3xh^2+h^3+2h}{h}\\\\\\\frac{h.\left(3x^2+3xh+h^2+2\right)}{h}\\\\\\\boxed{3x^2+3xh+h^2+2}$

2)

$\frac{\frac{1}{x+h}-\frac{1}{x}}{h}\\\\\\\frac{\frac{1.(x)-1.(x+h)}{(x+h)x}}{h}\\\\\\\frac{\frac{x-x-h}{(x+h)x}}{h}\\\\\\\frac{-h}{x.(x+h).h}\\\\\\\frac{-1}{x.(x+h)}\\\\\\\boxed{-\frac{1}{x.(x+h)}}$