f(x) = raiz(x)

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

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

Tranquilo! :)

Resposta

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