a. \(y' = \left( {\dfrac{{x + 1}}{2}} \right)'.\dfrac{1}{{{{\cos }^2}\dfrac{{x + 1}}{2}}}\) \(\displaystyle = {1 \over {2{{\cos }^2}{{x + 1} \over 2}}}\)
b. \(y' = \left( {\sqrt {{x^2} + 1} } \right)'.\dfrac{{ - 1}}{{{{\sin }^2}\sqrt {{x^2} + 1} }}\)\( = \left( {{x^2} + 1} \right)'.\dfrac{1}{{2\sqrt {{x^2} + 1} }}.\dfrac{{ - 1}}{{{{\sin }^2}\sqrt {{x^2} + 1} }}\) \( = \dfrac{{ - 2x}}{{2\sqrt {{x^2} + 1} }}.\dfrac{1}{{{{\sin }^2}\sqrt {{x^2} + 1} }}\)
\(\displaystyle = {{ - x} \over {\sqrt {{x^2} + 1} }}.{1 \over {{{\sin }^2}\sqrt {{x^2} + 1} }}\)
c. \(y' = 3{\tan ^2}x\left( {\tan x} \right)' + \left( {2x} \right)'.\dfrac{{ - 1}}{{{{\sin }^2}2x}}\) \( = 3{\tan ^2}x.\dfrac{1}{{{{\cos }^2}x}} - \dfrac{2}{{{{\sin }^2}2x}}\) \(\displaystyle = {{3{{\tan }^2}x} \over {{{\cos }^2}x}} - {2 \over {{{\sin }^2}2x}}\)
d. \(y' = \left( {3x} \right)'.\dfrac{1}{{{{\cos }^2}3x}} - \left( {3x} \right)'.\dfrac{{ - 1}}{{{{\sin }^2}3x}}\) \(\displaystyle = {3 \over {{{\cos }^2}3x}} + {3 \over {{{\sin }^2}3x}} = {{12} \over {{{\sin }^2}6x}}\)
e. \(y' = \left( {1 + 2\tan x} \right)'.\dfrac{1}{{2\sqrt {1 + 2\tan x} }}\) \( = 2\left( {\tan x} \right)'.\dfrac{1}{{2\sqrt {1 + 2\tan x} }}\) \( = \dfrac{1}{{{{\cos }^2}x}}.\dfrac{1}{{\sqrt {1 + 2\tan x} }}\) \(\displaystyle = {1 \over {{\sqrt {1 + 2\tan x}.{\cos }^2}x }}\)
f. \(y' = x'\cot x + x.\left( {\cot x} \right)'\) \( = \cot x + x.\dfrac{{ - 1}}{{{{\sin }^2}x}}\) \(\displaystyle = \cot x - {x \over {{{\sin }^2}x}}\)
