Caracterize a função derivada

$$f(x) = {x^2}\operatorname{sen} x$$

$${D_f} = \mathbb{R}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {{x^2}\operatorname{sen} x} \right)}^\prime }} \\
{}& = &{{{\left( {{x^2}} \right)}^\prime }\operatorname{sen} x + {x^2}{{\left( {\operatorname{sen} x} \right)}^\prime }} \\
{}& = &{2x\operatorname{sen} x + {x^2}\cos x}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\mathbb{R} \to \mathbb{R}} \\
{}&{x \to 2x\operatorname{sen} x + {x^2}\cos x}
\end{array}$$

$$f(x) = 5x\cos \left( {3x} \right)$$

$${D_f} = \mathbb{R}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {5x\cos \left( {3x} \right)} \right)}^\prime }} \\
{}& = &{{{\left( {5x} \right)}^\prime }\cos \left( {3x} \right) + 5x{{\left( {\cos \left( {3x} \right)} \right)}^\prime }} \\
{}& = &{5\cos \left( {3x} \right) – 15x\operatorname{sen} \left( {3x} \right)}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\mathbb{R} \to \mathbb{R}} \\
{}&{x \to 5\cos \left( {3x} \right) – 15x\operatorname{sen} \left( {3x} \right)}
\end{array}$$

$$f(x) = \frac{{1 – \cos x}}{{1 + \cos x}}$$

$${D_f} = \left\{ {x \in \mathbb{R}:1 + \cos x \ne 0} \right\} = \left\{ {x \in \mathbb{R}:x \ne \pi  + 2k\pi ,k \in \mathbb{Z}} \right\}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {\frac{{1 – \cos x}}{{1 + \cos x}}} \right)}^\prime }} \\
{}& = &{\frac{{{{\left( {1 – \cos x} \right)}^\prime } \times \left( {1 + \cos x} \right) – {{\left( {1 + \cos x} \right)}^\prime } \times \left( {1 – \cos x} \right)}}{{{{\left( {1 + \cos x} \right)}^2}}}} \\
{}& = &{\frac{{\operatorname{sen} x \times \left( {1 + \cos x} \right) + \operatorname{sen} x \times \left( {1 – \cos x} \right)}}{{{{\left( {1 + \cos x} \right)}^2}}}} \\
{}& = &{\frac{{2\operatorname{sen} x}}{{{{\left( {1 + \cos x} \right)}^2}}}}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\left\{ {x \in \mathbb{R}:x \ne \pi  + 2k\pi ,k \in \mathbb{Z}} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{2\operatorname{sen} x}}{{{{\left( {1 + \cos x} \right)}^2}}}}
\end{array}$$

$$f(x) = \frac{x}{{\operatorname{sen} x}}$$

$${D_f} = \left\{ {x \in \mathbb{R}:\operatorname{sen} x \ne 0} \right\} = \left\{ {x \in \mathbb{R}:x \ne k\pi ,k \in \mathbb{Z}} \right\}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {\frac{x}{{\operatorname{sen} x}}} \right)}^\prime }} \\
{}& = &{\frac{{\operatorname{sen} x – x\cos x}}{{{{\operatorname{sen} }^2}x}}}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\left\{ {x \in \mathbb{R}:x \ne k\pi ,k \in \mathbb{Z}} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{\operatorname{sen} x – x\cos x}}{{{{\operatorname{sen} }^2}x}}}
\end{array}$$

$$f(x) = \frac{{\operatorname{tg} x}}{{1 + {x^2}}}$$

$${D_f} = \left\{ {x \in \mathbb{R}:x \ne \frac{\pi }{2} + k\pi ,k \in \mathbb{Z}} \right\}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {\frac{{\operatorname{tg} x}}{{1 + {x^2}}}} \right)}^\prime }} \\
{}& = &{\frac{{\frac{1}{{{{\cos }^2}x}} \times \left( {1 + {x^2}} \right) – 2x\operatorname{tg} x}}{{{{\left( {1 + {x^2}} \right)}^2}}}} \\
{}& = &{\frac{{\left( {1 + {{\operatorname{tg} }^2}x} \right) \times \left( {1 + {x^2}} \right) – 2x\operatorname{tg} x}}{{{{\left( {1 + {x^2}} \right)}^2}}}} \\
{}& = &{\frac{{1 + {x^2} + {{\operatorname{tg} }^2}x + {x^2}{{\operatorname{tg} }^2}x – 2x\operatorname{tg} x}}{{{{\left( {1 + {x^2}} \right)}^2}}}}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\left\{ {x \in \mathbb{R}:x \ne \frac{\pi }{2} + k\pi ,k \in \mathbb{Z}} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{1 + {x^2} + {{\operatorname{tg} }^2}x + {x^2}{{\operatorname{tg} }^2}x – 2x\operatorname{tg} x}}{{{{\left( {1 + {x^2}} \right)}^2}}}}
\end{array}$$

$$f(x) = \frac{{1 – \cos \left( {2x} \right)}}{{2x}}$$

$${D_f} = \mathbb{R}\backslash \left\{ 0 \right\}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {\frac{{1 – \cos \left( {2x} \right)}}{{2x}}} \right)}^\prime }} \\
{}& = &{\frac{{2\operatorname{sen} \left( {2x} \right) \times 2x – 2\left( {1 – \cos \left( {2x} \right)} \right)}}{{{{\left( {2x} \right)}^2}}}} \\
{}& = &{\frac{{4x\operatorname{sen} \left( {2x} \right) – 2 + 2\cos \left( {2x} \right)}}{{4{x^2}}}} \\
{}& = &{\frac{{2x\operatorname{sen} \left( {2x} \right) – 1 + \cos \left( {2x} \right)}}{{2{x^2}}}}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{2x\operatorname{sen} \left( {2x} \right) – 1 + \cos \left( {2x} \right)}}{{2{x^2}}}}
\end{array}$$

$$f(x) = {\left( {\cos x + \operatorname{sen} x} \right)^2}$$

$${D_f} = \mathbb{R}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {{{\left( {\cos x + \operatorname{sen} x} \right)}^2}} \right)}^\prime }} \\
{}& = &{2\left( {\cos x + \operatorname{sen} x} \right){{\left( {\cos x + \operatorname{sen} x} \right)}^\prime }} \\
{}& = &{2\left( {\cos x + \operatorname{sen} x} \right)\left( { – \operatorname{sen} x + \cos x} \right)} \\
{}& = &{2\left( {{{\cos }^2}x – {{\operatorname{sen} }^2}x} \right)} \\
{}& = &{2\cos \left( {2x} \right)}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\mathbb{R} \to \mathbb{R}} \\
{}&{x \to 2\cos \left( {2x} \right)}
\end{array}$$

$$f(x) = \frac{1}{{\operatorname{sen} x\cos x}}$$

$${D_f} = \left\{ {x \in \mathbb{R}:\operatorname{sen} x\cos x \ne 0} \right\} = \left\{ {x \in \mathbb{R}:x \ne \frac{{k\pi }}{2},k \in \mathbb{Z}} \right\}$$
\[\begin{array}{*{20}{l}}
{f’\left( x \right)}& = &{{{\left( {\frac{1}{{\operatorname{sen} x\cos x}}} \right)}^\prime }} \\
{}& = &{\frac{{ – {{\left( {\operatorname{sen} x\cos x} \right)}^\prime }}}{{{{\left( {\operatorname{sen} x\cos x} \right)}^2}}}} \\
{}& = &{\frac{{ – \left( {{{\cos }^2}x – {{\operatorname{sen} }^2}x} \right)}}{{{{\left( {\operatorname{sen} x\cos x} \right)}^2}}}} \\
{}& = &{\frac{{ – \cos \left( {2x} \right)}}{{{{\left( {\operatorname{sen} x\cos x} \right)}^2}}}} \\
{}& = &{\frac{{ – 4\cos \left( {2x} \right)}}{{{{\left( {2\operatorname{sen} x\cos x} \right)}^2}}}} \\
{}& = &{\frac{{ – 4\cos \left( {2x} \right)}}{{{{\operatorname{sen} }^2}\left( {2x} \right)}}}
\end{array}\]
$$\begin{array}{*{20}{l}}
{f’:}&{\left\{ {x \in \mathbb{R}:x \ne \frac{{k\pi }}{2},k \in \mathbb{Z}} \right\} \to \mathbb{R}} \\
{}&{x \to \frac{{ – 4\cos \left( {2x} \right)}}{{{{\operatorname{sen} }^2}\left( {2x} \right)}}}
\end{array}$$