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

y = 1 / raiz(x - 1)

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)) - (1)/(sqrt(x - 1)))/(Delta x)

Agora vamos manipular o limite:

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

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

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

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

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

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 - 1) sqrt(x - 1) (sqrt(x - 1) + sqrt(x + Delta x - 1))) =

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

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

= (- 1)/((x - 1) (2 sqrt(x - 1))) = (- 1)/(2 (x - 1)^((3)/(2)))

Resposta

(d y)/(d x) = (- 1)/(2 (x - 1)^((3)/(2)))

Ver Outros Exercícios desse livro

Exercícios de Livros Relacionados