Bài tập 31 trang 212 SGK Toán 11 NC

a)

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

b)

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

c)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
y\prime  = 3{\tan ^2}x(\tan x)\prime  + (2x)\prime .\frac{{ - 1}}{{{{\sin }^2}2x}}\\
 = 3{\tan ^2}x.\frac{1}{{{{\cos }^2}x}} - \frac{2}{{{{\sin }^2}2x}}
\end{array}\\
{ = \frac{{3{{\tan }^2}x}}{{{{\cos }^2}x}} - \frac{2}{{{{\sin }^2}2x}}}
\end{array}\)

d)

\(\begin{array}{l}
y\prime  = (3x)\prime .\frac{1}{{{{\cos }^2}3x}} - (3x)\prime .\frac{{ - 1}}{{{{\sin }^2}3x}}\\
 = \frac{3}{{{{\cos }^2}3x}} + \frac{3}{{{{\sin }^2}3x}} = \frac{{12}}{{{{\sin }^2}6x}}
\end{array}\)

e)

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

f)

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

-- Mod Toán 11 HỌC247