Derive.
17. f(w)=1+sec w1-sec w

Passo 1

O enunciado pede a derivada de
f(w)=1+sec w1-sec w
Aqui vamos usar a regra do quociente e a derivada de sec w, que é
ddw[sec w]=sec wtan w

Passo 2

Definimos o numerador N(w) e o denominador D(w) de f:
N(w)=1+sec w, D(w)=1-sec w
As derivadas são
N'(w)=ddw[1+sec w]=sec wtan w, D'(w)=ddw[1-sec w]=-sec wtan w

Passo 3

Aplicando a regra do quociente
f'(w) =N'(w),D(w)-N(w),D'(w)[D(w)]^2,
temos
f'(w) =l(sec wtan wr),(1-sec w);-;(1+sec w),l(-sec wtan wr)(1-sec w)^2

Passo 4

No numerador, fatoramos sec wtan w:
l(sec wtan wr),(1-sec w)+(1+sec w),l(sec wtan wr) =sec wtan wl[(1-sec w)+(1+sec w)r] =2sec wtan w
Portanto
f'(w) =2sec wtan w(1-sec w)^2

Resposta

f'(w)=2sec wtan w(1-sec w)^2

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