Calcule a derivada.

g(x) = e^(sec(raiz(x))) / x

Passo 1

Aqui de cara dá pra ver que vamos usar a regra da divisão:

f^' (x) = (h^' (x) t (x) - h (x) t ' (x))/([t (x)]^2)

Onde:

h (x) = e^(sec sqrt(x))

e

t (x) = e^t + e^(- t)

Assim temos então que:

((e^(sec sqrt(x)))/(x))^' = (h^' (x) t (x) - h (x) t ' (x))/([t (x)]^2) = ((e^(sec sqrt(x)))^' (x) - (e^(sec sqrt(x))) (x)^')/([x]^2)

Passo 2

Vamos ver as regras que vamos precisar para calcular as derivadas, vamos usar a regra da função exponencial, da função trigonométrica e do polinômio:

(e^(f (x)))^' = e^(f (x)) * f ( x ) '

(sec (x))^' = sec (x) * tg (x)

(x^n)^' = n x^(n - 1)

Passo 3

Primeiro vamos calcular h^' (x) :

h^' (x) = (e^(sec sqrt(x)))^' = e^(sec sqrt(x)) (sec sqrt(x))^' = e^(sec sqrt(x)) * sec sqrt(x) * tg sqrt(x) * (sqrt(x))^' = e^(sec sqrt(x)) * sec sqrt(x) * tg sqrt(x) * ((1)/(2) x^((1)/(2) - 1)) = e^(sec sqrt(x)) * sec sqrt(x) * tg sqrt(x) * ((1)/(2) x^(- (1)/(2))) = (e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x))/(2 sqrt(x))

Agora para t^' (x) :

t^' (x) = (x)^' = x^(1 - 1) = x^0 = 1

Passo 4

Pronto, agora é só substituir na expressão:

g^' (x) = ((e^(sec sqrt(x)))/(x))^' = ((e^(sec sqrt(x)))^' (x) - (e^(sec sqrt(x))) (x)^')/([x]^2) = (((e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x))/(2 sqrt(x))) (x) - (e^(sec sqrt(x))) (1))/(x^2) = ((x e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x))/(2 sqrt(x)) - e^(sec sqrt(x)))/(x^2) = ((x e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x) - 2 sqrt(x) e^(sec sqrt(x)))/(2 sqrt(x)))/(x^2) = (x e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x) - 2 sqrt(x) e^(sec sqrt(x)))/(2 sqrt(x)) * ((1)/(x^2)) = (x e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x) - 2 sqrt(x) e^(sec sqrt(x)))/(2 x^2 sqrt(x))

Resposta

g^' (x) = (x e^(sec sqrt(x)) sec sqrt(x) tg sqrt(x) - 2 sqrt(x) e^(sec sqrt(x)))/(2 x^2 sqrt(x))

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