Exercício 9: (Derivada implícita) Determine a derivada:
x^{2}\left[3 \operatorname{tg} 4 x+4 x \operatorname{sec} ^{2} 4 x\right]

Passo 1

O enunciado pede para calcular a derivada da função
f(x) = x^{2}\bigl[\,3\tan(4x)+4x\operatorname{sec}^{2}(4x)\bigr].
O conteúdo envolvido é a regra do produto, a regra da cadeia e as derivadas \frac{d}{du}\tan(u)=\operatorname{sec}^{2}(u) e \frac{d}{du}\operatorname{sec}(u)=\operatorname{sec}(u)\tan(u).

Passo 2

Vamos escrever
f(x)=x^{2}\,g(x),
onde
g(x)=3\tan(4x)+4x\operatorname{sec}^{2}(4x).
Pela regra do produto,
f'(x)=2x\,g(x)+x^{2}\,g'(x).

Passo 3

Agora calculamos g'(x). Como
g(x)=3\tan(4x)+4x\operatorname{sec}^{2}(4x),
temos duas parcelas:
  1. Derivada de 3\tan(4x):
    pela regra da cadeia,
\frac{d}{dx}\bigl[3\tan(4x)\bigr] =3\bigl[\operatorname{sec}^{2}(4x)\bigr]\cdot(4) =12\operatorname{sec}^{2}(4x).
  1. Derivada de 4x\operatorname{sec}^{2}(4x):
    aplicando produto u=4x, v=\operatorname{sec}^{2}(4x),
\frac{d}{dx}(4x\operatorname{sec}^{2}(4x)) =4\operatorname{sec}^{2}(4x)+4x\cdot\frac{d}{dx}\bigl[\operatorname{sec}^{2}(4x)\bigr].
E
\frac{d}{dx}\bigl[\operatorname{sec}^{2}(4x)\bigr] =2\operatorname{sec}^{2}(4x)\tan(4x)\cdot(4) =8\operatorname{sec}^{2}(4x)\tan(4x).
Logo,
\frac{d}{dx}(4x\operatorname{sec}^{2}(4x)) =4\operatorname{sec}^{2}(4x)+4x\bigl[8\operatorname{sec}^{2}(4x)\tan(4x)\bigr] =4\operatorname{sec}^{2}(4x)+32x\operatorname{sec}^{2}(4x)\tan(4x).
Somando as duas parcelas, obtemos
g'(x)=12\operatorname{sec}^{2}(4x)+4\operatorname{sec}^{2}(4x)+32x\operatorname{sec}^{2}(4x)\tan(4x) =16\operatorname{sec}^{2}(4x)+32x\operatorname{sec}^{2}(4x)\tan(4x).

Passo 4

Substituindo g(x) e g'(x) em
f'(x)=2x\,g(x)+x^{2}\,g'(x),
temos
f'(x) =2x\bigl[3\tan(4x)+4x\operatorname{sec}^{2}(4x)\bigr] +x^{2}\bigl[16\operatorname{sec}^{2}(4x)+32x\operatorname{sec}^{2}(4x)\tan(4x)\bigr].
Distribuindo e somando termos semelhantes:
f'(x) =6x\tan(4x)+8x^{2}\operatorname{sec}^{2}(4x) +16x^{2}\operatorname{sec}^{2}(4x)+32x^{3}\operatorname{sec}^{2}(4x)\tan(4x),
f'(x) =6x\tan(4x)+24x^{2}\operatorname{sec}^{2}(4x)+32x^{3}\operatorname{sec}^{2}(4x)\tan(4x).

Resposta

f'(x)=6x\tan(4x)+24x^{2}\operatorname{sec}^{2}(4x)+32x^{3}\operatorname{sec}^{2}(4x)\tan(4x).

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