Use a Fórmula (12) para encontrar dy / dx .

y = 1 / raiz(x)

Passo 1

Vamos usar a Fórmula 1 2 para encontrar a derivada:

(d y)/(d x) = underset(Delta x -> 0)(lim) (Delta y)/(Delta x) = underset(Delta x -> 0)(lim) (f (x + Delta x) - f (x))/(Delta x)

Passo 2

Vamos então aplicar:

(d y)/(d x) = underset(Delta x -> 0)(lim) (f (x + Delta x) - f (x))/(Delta x) = underset(Delta x -> 0)(lim) ((1)/(sqrt(x + Delta x)) - (1)/(sqrt(x)))/(Delta x)

Agora vamos manipular o limite:

underset(Delta x -> 0)(lim) ((1)/(sqrt(x + Delta x)) - (1)/(sqrt(x)))/(Delta x) = underset(Delta x -> 0)(lim) ((sqrt(x) - sqrt(x + Delta x))/(sqrt(x + Delta x) sqrt(x)))/(Delta x) = underset(Delta x -> 0)(lim) (sqrt(x) - sqrt(x + Delta x))/(sqrt(x + Delta x) sqrt(x)) * (1)/(Delta x) = underset(Delta x -> 0)(lim) (sqrt(x) - sqrt(x + Delta x))/(Delta x sqrt(x + Delta x) sqrt(x)) =

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

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

E por último vamos cancelar o termo Delta x do numerador e do denominador e vamos aplicar o limite:

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

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

Resposta

(d y)/(d x) = (- 1)/(2 x sqrt(x))

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