Recorrendo \u00e0s regras de deriva\u00e7\u00e3o, caracterize a fun\u00e7\u00e3o derivada em cada um dos casos seguintes:<\/p>\n
\n$$f(x) = {x^2}\\operatorname{sen} x$$<\/p>\n<\/blockquote>\n
$${D_f} = \\mathbb{R}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {{x^2}\\operatorname{sen} x} \\right)}^\\prime }} \\\\
\n{}& = &{{{\\left( {{x^2}} \\right)}^\\prime }\\operatorname{sen} x + {x^2}{{\\left( {\\operatorname{sen} x} \\right)}^\\prime }} \\\\
\n{}& = &{2x\\operatorname{sen} x + {x^2}\\cos x}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\mathbb{R} \\to \\mathbb{R}} \\\\
\n{}&{x \\to 2x\\operatorname{sen} x + {x^2}\\cos x}
\n\\end{array}$$<\/p>\n<< Enunciado<\/a><\/span>R2 >><\/a><\/span><\/div><\/div>\n\n\nR2<\/b><\/span><\/p>\n\n$$f(x) = 5x\\cos \\left( {3x} \\right)$$<\/p>\n<\/blockquote>\n
$${D_f} = \\mathbb{R}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {5x\\cos \\left( {3x} \\right)} \\right)}^\\prime }} \\\\
\n{}& = &{{{\\left( {5x} \\right)}^\\prime }\\cos \\left( {3x} \\right) + 5x{{\\left( {\\cos \\left( {3x} \\right)} \\right)}^\\prime }} \\\\
\n{}& = &{5\\cos \\left( {3x} \\right) – 15x\\operatorname{sen} \\left( {3x} \\right)}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\mathbb{R} \\to \\mathbb{R}} \\\\
\n{}&{x \\to 5\\cos \\left( {3x} \\right) – 15x\\operatorname{sen} \\left( {3x} \\right)}
\n\\end{array}$$<\/p>\n<< R1<\/a><\/span>R3 >><\/a><\/span><\/div><\/div>\n\n\nR3<\/b><\/span><\/p>\n\n$$f(x) = \\frac{{1 – \\cos x}}{{1 + \\cos x}}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\left\\{ {x \\in \\mathbb{R}:1 + \\cos x \\ne 0} \\right\\} = \\left\\{ {x \\in \\mathbb{R}:x \\ne \\pi\u00a0 + 2k\\pi ,k \\in \\mathbb{Z}} \\right\\}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {\\frac{{1 – \\cos x}}{{1 + \\cos x}}} \\right)}^\\prime }} \\\\
\n{}& = &{\\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}}}} \\\\
\n{}& = &{\\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}}}} \\\\
\n{}& = &{\\frac{{2\\operatorname{sen} x}}{{{{\\left( {1 + \\cos x} \\right)}^2}}}}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\left\\{ {x \\in \\mathbb{R}:x \\ne \\pi\u00a0 + 2k\\pi ,k \\in \\mathbb{Z}} \\right\\} \\to \\mathbb{R}} \\\\
\n{}&{x \\to \\frac{{2\\operatorname{sen} x}}{{{{\\left( {1 + \\cos x} \\right)}^2}}}}
\n\\end{array}$$<\/p>\n<< R2<\/a><\/span>R4 >><\/a><\/span><\/div><\/div>\n\n\nR4<\/b><\/span><\/p>\n\n$$f(x) = \\frac{x}{{\\operatorname{sen} x}}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\left\\{ {x \\in \\mathbb{R}:\\operatorname{sen} x \\ne 0} \\right\\} = \\left\\{ {x \\in \\mathbb{R}:x \\ne k\\pi ,k \\in \\mathbb{Z}} \\right\\}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {\\frac{x}{{\\operatorname{sen} x}}} \\right)}^\\prime }} \\\\
\n{}& = &{\\frac{{\\operatorname{sen} x – x\\cos x}}{{{{\\operatorname{sen} }^2}x}}}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\left\\{ {x \\in \\mathbb{R}:x \\ne k\\pi ,k \\in \\mathbb{Z}} \\right\\} \\to \\mathbb{R}} \\\\
\n{}&{x \\to \\frac{{\\operatorname{sen} x – x\\cos x}}{{{{\\operatorname{sen} }^2}x}}}
\n\\end{array}$$<\/p>\n<< R3<\/a><\/span>R5 >><\/a><\/span><\/div><\/div>\n\n\nR5<\/b><\/span><\/p>\n\n$$f(x) = \\frac{{\\operatorname{tg} x}}{{1 + {x^2}}}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{\\pi }{2} + k\\pi ,k \\in \\mathbb{Z}} \\right\\}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {\\frac{{\\operatorname{tg} x}}{{1 + {x^2}}}} \\right)}^\\prime }} \\\\
\n{}& = &{\\frac{{\\frac{1}{{{{\\cos }^2}x}} \\times \\left( {1 + {x^2}} \\right) – 2x\\operatorname{tg} x}}{{{{\\left( {1 + {x^2}} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{\\left( {1 + {{\\operatorname{tg} }^2}x} \\right) \\times \\left( {1 + {x^2}} \\right) – 2x\\operatorname{tg} x}}{{{{\\left( {1 + {x^2}} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{1 + {x^2} + {{\\operatorname{tg} }^2}x + {x^2}{{\\operatorname{tg} }^2}x – 2x\\operatorname{tg} x}}{{{{\\left( {1 + {x^2}} \\right)}^2}}}}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{\\pi }{2} + k\\pi ,k \\in \\mathbb{Z}} \\right\\} \\to \\mathbb{R}} \\\\
\n{}&{x \\to \\frac{{1 + {x^2} + {{\\operatorname{tg} }^2}x + {x^2}{{\\operatorname{tg} }^2}x – 2x\\operatorname{tg} x}}{{{{\\left( {1 + {x^2}} \\right)}^2}}}}
\n\\end{array}$$<\/p>\n<< R4<\/a><\/span>R6 >><\/a><\/span><\/div><\/div>\n\n\nR6<\/b><\/span><\/p>\n\n$$f(x) = \\frac{{1 – \\cos \\left( {2x} \\right)}}{{2x}}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\mathbb{R}\\backslash \\left\\{ 0 \\right\\}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {\\frac{{1 – \\cos \\left( {2x} \\right)}}{{2x}}} \\right)}^\\prime }} \\\\
\n{}& = &{\\frac{{2\\operatorname{sen} \\left( {2x} \\right) \\times 2x – 2\\left( {1 – \\cos \\left( {2x} \\right)} \\right)}}{{{{\\left( {2x} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{4x\\operatorname{sen} \\left( {2x} \\right) – 2 + 2\\cos \\left( {2x} \\right)}}{{4{x^2}}}} \\\\
\n{}& = &{\\frac{{2x\\operatorname{sen} \\left( {2x} \\right) – 1 + \\cos \\left( {2x} \\right)}}{{2{x^2}}}}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\mathbb{R}\\backslash \\left\\{ 0 \\right\\} \\to \\mathbb{R}} \\\\
\n{}&{x \\to \\frac{{2x\\operatorname{sen} \\left( {2x} \\right) – 1 + \\cos \\left( {2x} \\right)}}{{2{x^2}}}}
\n\\end{array}$$<\/p>\n<< R5<\/a><\/span>R7 >><\/a><\/span><\/div><\/div>\n\n\nR7<\/b><\/span><\/p>\n\n$$f(x) = {\\left( {\\cos x + \\operatorname{sen} x} \\right)^2}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\mathbb{R}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {{{\\left( {\\cos x + \\operatorname{sen} x} \\right)}^2}} \\right)}^\\prime }} \\\\
\n{}& = &{2\\left( {\\cos x + \\operatorname{sen} x} \\right){{\\left( {\\cos x + \\operatorname{sen} x} \\right)}^\\prime }} \\\\
\n{}& = &{2\\left( {\\cos x + \\operatorname{sen} x} \\right)\\left( { – \\operatorname{sen} x + \\cos x} \\right)} \\\\
\n{}& = &{2\\left( {{{\\cos }^2}x – {{\\operatorname{sen} }^2}x} \\right)} \\\\
\n{}& = &{2\\cos \\left( {2x} \\right)}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\mathbb{R} \\to \\mathbb{R}} \\\\
\n{}&{x \\to 2\\cos \\left( {2x} \\right)}
\n\\end{array}$$<\/p>\n<< R6<\/a><\/span>R8 >><\/a><\/span><\/div><\/div>\n\n\nR8<\/b><\/span><\/p>\n\n$$f(x) = \\frac{1}{{\\operatorname{sen} x\\cos x}}$$<\/p>\n<\/blockquote>\n
\n$${D_f} = \\left\\{ {x \\in \\mathbb{R}:\\operatorname{sen} x\\cos x \\ne 0} \\right\\} = \\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{{k\\pi }}{2},k \\in \\mathbb{Z}} \\right\\}$$
\n\\[\\begin{array}{*{20}{l}}
\n{f’\\left( x \\right)}& = &{{{\\left( {\\frac{1}{{\\operatorname{sen} x\\cos x}}} \\right)}^\\prime }} \\\\
\n{}& = &{\\frac{{ – {{\\left( {\\operatorname{sen} x\\cos x} \\right)}^\\prime }}}{{{{\\left( {\\operatorname{sen} x\\cos x} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{ – \\left( {{{\\cos }^2}x – {{\\operatorname{sen} }^2}x} \\right)}}{{{{\\left( {\\operatorname{sen} x\\cos x} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{ – \\cos \\left( {2x} \\right)}}{{{{\\left( {\\operatorname{sen} x\\cos x} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{ – 4\\cos \\left( {2x} \\right)}}{{{{\\left( {2\\operatorname{sen} x\\cos x} \\right)}^2}}}} \\\\
\n{}& = &{\\frac{{ – 4\\cos \\left( {2x} \\right)}}{{{{\\operatorname{sen} }^2}\\left( {2x} \\right)}}}
\n\\end{array}\\]
\n$$\\begin{array}{*{20}{l}}
\n{f’:}&{\\left\\{ {x \\in \\mathbb{R}:x \\ne \\frac{{k\\pi }}{2},k \\in \\mathbb{Z}} \\right\\} \\to \\mathbb{R}} \\\\
\n{}&{x \\to \\frac{{ – 4\\cos \\left( {2x} \\right)}}{{{{\\operatorname{sen} }^2}\\left( {2x} \\right)}}}
\n\\end{array}$$<\/p>\n