Derivar e Siuplificar
Passo 1
O enunciado pede a derivada de
. Vamos usar regra da cadeia para logaritmo natural e derivada de tangente, além da regra do quociente para funções trigonométricas.
Passo 2
Definimos
e calculamos
pela regra do quociente. Seja
as derivadas são
$$N’ = \tfrac{1}{2}\sec^2\bigl(\tfrac{x}{2}\bigr),\quad D’ = -\tfrac{1}{2}\sec^2\bigl(\tfrac{x}{2}\bigr).
$$
Passo 3
Então
$$u’ = \frac{N’D - N D’}{D^2} = \frac{\tfrac12\sec^2(\tfrac{x}{2})(1-\tan(\tfrac{x}{2})) - (1+\tan(\tfrac{x}{2}))\bigl(-\tfrac12\sec2(\tfrac{x}{2})\bigr)}{(1-\tan(\tfrac{x}{2}))2} = \frac{\sec2(\tfrac{x}{2})}{(1-\tan(\tfrac{x}{2}))2}.
$$
Passo 4
Agora derivamos
usando
:
$$y’ = \frac{u’}{u} = \frac{\tfrac{\sec2(\tfrac{x}{2})}{(1-\tan(\tfrac{x}{2}))2}}{\tfrac{1+\tan(\tfrac{x}{2})}{1-\tan(\tfrac{x}{2})}} = \frac{\sec^2(\tfrac{x}{2})}{(1-\tan(\tfrac{x}{2}))(1+\tan(\tfrac{x}{2}))} = \frac{\sec2(\tfrac{x}{2})}{1-\tan2(\tfrac{x}{2})}.
$$
Passo 5
Para simplificar usamos identidades trigonométricas:
e também
$$y’ = \frac{1/\cos2(\tfrac{x}{2})}{\cos(x)/\cos2(\tfrac{x}{2})} = \frac{1}{\cos(x)} = \sec(x).
$$
Resposta
y’ =