Bài tập 33 trang 212 SGK Toán 11 NC

a)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
y' = \frac{{{{\left( {\sin x} \right)}^\prime }.x - \sin x.\left( {x'} \right)}}{{{x^2}}}\\
 + \frac{{x\prime \sin x - x.(\sin x)\prime }}{{{{\sin }^2}x}}
\end{array}\\
{ = \frac{{x\cos x - \sin x}}{{{x^2}}} + \frac{{\sin x - x\cos x}}{{{{\sin }^2}x}}}\\
{ = (x\cos x - \sin x)\left( {\frac{1}{{{x^2}}} - \frac{1}{{{{\sin }^2}x}}} \right)}
\end{array}\)

b)

\(\begin{array}{l}
y' = \frac{{{{\left( {{{\sin }^2}x} \right)}^\prime }.\left( {1 + \tan 2x} \right) - {{\sin }^2}x\left( {1 + \tan 2x} \right)'}}{{{{\left( {1 + \tan 2x} \right)}^2}}}\\
 = \frac{{2\sin x\left( {sinx} \right)'\left( {1 + \tan 2x} \right) - {{\sin }^2}x\left( {2x} \right)'.\left( {1 + {{\tan }^2}2x} \right)}}{{{{\left( {1 + {{\tan }^2}2x} \right)}^2}}}\\
 = \frac{{2\sin x\cos x\left( {1 + \tan 2x} \right) - {{\sin }^2}x.2\left( {1 + {{\tan }^2}2x} \right)}}{{{{\left( {1 + \tan 2x} \right)}^2}}}\\
 = \frac{{\sin 2x}}{{1 + \tan 2x}} - \frac{{2{{\sin }^2}x\left( {1 + {{\tan }^2}2x} \right)}}{{{{\left( {1 + \tan 2x} \right)}^2}}}
\end{array}\)

c)

\(y' = {\left( {\sin x} \right)^\prime }.\frac{1}{{{{\cos }^2}\left( {\sin x} \right)}} = \frac{{\cos x}}{{{{\cos }^2}(\sin x)}}\)

d)

\(\begin{array}{*{20}{l}}
{y' = x'.\cot \left( {{x^2} - 1} \right) + x.{{\left[ {\cot \left( {{x^2} - 1} \right)} \right]}^\prime }}\\
{ = \cot ({x^2} - 1) + x.({x^2} - 1)\prime .\frac{{ - 1}}{{\sin 2({x^2} - 1)}}}\\
{ = \cot ({x^2} - 1) + x.\frac{{ - 2x}}{{{{\sin }^2}({x^2} - 1)}}}\\
{ = \cot ({x^2} - 1) - \frac{{2{x^2}}}{{{{\sin }^2}({x^2} - 1)}}}
\end{array}\)

e)

\(\begin{array}{l}
y' = 2{\left( {\cos \sqrt {\frac{\pi }{4} - 2x} } \right)^\prime }.\cos \sqrt {\frac{\pi }{4} - 2x} \\
 = 2\left( {\sqrt {\frac{\pi }{4} - 2x} } \right)'.\left( { - \sin \sqrt {\frac{\pi }{4} - 2x} } \right).\cos \sqrt {\frac{\pi }{4} - 2x} \\
 = \frac{{\left( {\frac{\pi }{4} - 2x} \right)'}}{{2.\sqrt {\frac{\pi }{4} - 2x} }}.\left( { - 2\sin \sqrt {\frac{\pi }{4} - 2x} .\cos \sqrt {\frac{\pi }{4} - 2x} } \right)\\
 = \frac{{ - 2}}{{2\sqrt {\frac{\pi }{4} - 2x} }}.\left( { - \sin 2\sqrt {\frac{\pi }{4} - 2x} } \right) = \frac{{2\sin \sqrt {\pi  - 8x} }}{{\sqrt {\pi  - 8x} }}
\end{array}\)

f)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
y' = x'\sqrt {\sin 3x}  + x.{\left( {\sqrt {\sin 3x} } \right)^\prime }\\
 = \sqrt {\sin 3x}  + x.\frac{{(\sin 3x)\prime }}{{2\sqrt {\sin 3x} }}
\end{array}\\
\begin{array}{l}
 = \sqrt {\sin 3x}  + x.\frac{{3\cos 3x}}{{2\sqrt {\sin 3x} }}\\
 = \frac{{2\sin 3x + 3x\cos 3x}}{{2\sqrt {\sin 3x} }}
\end{array}
\end{array}\)

-- Mod Toán 11 HỌC247