Calcule a derivada de cada uma das funções
Funções seno, co-seno e tangente: Infinito 12 A - Parte 3 Pág. 49 Ex. 22
Enunciado
Calcule a derivada de cada uma das funções reais de variável real:
- $f:x \to 3 + 2\cos x$
- $g:x \to \operatorname{sen} x + \cos x$
- $h:t \to \operatorname{sen} t.\cos t$
- $i:z \to 3z\cos z$
- $j:x \to 3x\operatorname{tg} x$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
{f'(x)}& = &{\left( {3 + 2\cos x} \right)’} \\
{}& = &{2 \times \left( {\cos x} \right)’} \\
{}& = &{ – 2\operatorname{sen} x}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{g'(x)}& = &{\left( {\operatorname{sen} x + \cos x} \right)’} \\
{}& = &{\left( {\operatorname{sen} x} \right)’ + \left( {\cos x} \right)’} \\
{}& = &{\cos x – \operatorname{sen} x}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{h'(t)}& = &{\left( {\operatorname{sen} t.\cos t} \right)’} \\
{}& = &{\left( {\operatorname{sen} t} \right)’ \times \cos t + \operatorname{sen} t \times \left( {\cos t} \right)’} \\
{}& = &{{{\cos }^2}t – {{\operatorname{sen} }^2}t}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{i'(z)}& = &{{{\left( {3z\cos z} \right)}^\prime }} \\
{}& = &{{{\left( {3z} \right)}^\prime } \times \cos z + 3z \times {{\left( {\cos z} \right)}^\prime }} \\
{}& = &{3\cos z – 3z\operatorname{sen} z}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{j'(x)}& = &{\left( {3x\operatorname{tg} x} \right)’} \\
{}& = &{\left( {3x} \right)’ \times \operatorname{tg} x + 3x \times \left( {\operatorname{tg} x} \right)’} \\
{}& = &{\begin{array}{*{20}{c}}
{3\operatorname{tg} x + \frac{{3x}}{{{{\cos }^2}x}},}&{x \ne \frac{\pi }{2} + k\pi ,k \in \mathbb{Z}}
\end{array}}
\end{array}$$
