f(x) = x^(-1/2)

Passo 1

Fala, pessoal!!!

Ele tá pedindo pra gente inicialmente calcular f^' (x) usando a definição de limite.

Lembrando:

f^' (x) = underset(h -> 0)(lim) (f (x + h) - f (x))/(h)

Passo 2

Para o nosso caso, temos:

f^' (x) = underset(h -> 0)(lim) ((x + h)^(- (1)/(2)) - x^(- (1)/(2)))/(h) = underset(h -> 0)(lim) ((1)/(sqrt(x + h)) - (1)/(sqrt(x)))/(h) = underset(h -> 0)(lim) ((sqrt(x) - sqrt(x + h))/(sqrt((x + h) x)))/(h)

= underset(h -> 0)(lim) (sqrt(x) - sqrt(x + h))/(h (sqrt(x^2 + h x))) = underset(h -> 0)(lim) ((sqrt(x) - sqrt(x + h)) (sqrt(x) + sqrt(x + h)))/(h (sqrt(x^2 + h x)) (sqrt(x) + sqrt(x + h))) = underset(h -> 0)(lim) (x - x - h)/(h (sqrt(x^2 + h x)) (sqrt(x) + sqrt(x + h)))

= underset(h -> 0)(lim) - (h)/(h (sqrt(x^2 + h x)) (sqrt(x) + sqrt(x + h))) = underset(h -> 0)(lim) - (1)/((sqrt(x^2 + h x)) (sqrt(x) + sqrt(x + h)))

= - (1)/((sqrt(x^2 + 0)) (sqrt(x) + sqrt(x + 0))) = - (1)/((sqrt(x^2)) (sqrt(x) + sqrt(x))) = - (1)/(x (2 sqrt(x))) = - (1)/(2 x sqrt(x))

Tranquilo! :)

Resposta

f^' (x) = - (1)/(2 x sqrt(x))