Tìm đạo hàm của mỗi hàm số sau :
a. \(y = {{\sin x} \over x} + {x \over {{\mathop{\rm sinx}\nolimits} }}\)
b. \(y = {{{{\sin }^2}x} \over {1 + \tan 2x}}\)
c. \(y = \tan \left( {\sin x} \right)\)
d. \(y = x\cot \left( {{x^2} - 1} \right)\)
e. \(y = {\cos ^2}\sqrt {{\pi \over 4} - 2x} \)
f. \(y = x\sqrt {\sin 3x} \)
a.
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\(\eqalign{ & y’ = {{x\cos x - \sin x} \over {{x^2}}} + {{\sin x - x\cos x} \over {{{\sin }^2}x}} \cr & = \left( {x\cos x - {\mathop{\rm sinx}\nolimits} } \right)\left( {{1 \over {{x^2}}} - {1 \over {{{\sin }^2}x}}} \right) \cr} \)
b.
\(\eqalign{ & y’ = {{2\sin x\cos x\left( {1 + \tan 2x} \right) - {{\sin }^2}x.2\left( {1 + {{\tan }^2}2x} \right)} \over {{{\left( {1 + \tan 2x} \right)}^2}}} \cr & = {{\sin 2x} \over {\left( {1 + \tan 2x} \right)}} - {{2{{\sin }^2}x\left( {1 + {{\tan }^2}2x} \right)} \over {{{\left( {1 + \tan 2x} \right)}^2}}} \cr} \)
c. \(y’ = {{\cos x} \over {{{\cos }^2}\left( {\sin x} \right)}}\)
d.
\(\eqalign{ & y’ = \cot \left( {{x^2} - 1} \right) + x.{{ - 2x} \over {{{\sin }^2}\left( {{x^2} - 1} \right)}} \cr & = \cot \left( {{x^2} - 1} \right) - {{2{x^2}} \over {{{\sin }^2}\left( {{x^2} - 1} \right)}} \cr} \)
e.
\(\eqalign{ & y = {1 \over 2}\left( {1 + \cos 2\sqrt {{\pi \over 4} - 2x} } \right) \cr & y’ = - {1 \over 2}. \sin 2\sqrt {{\pi \over 4} - 2x} .\,2{{ - 2} \over {2\sqrt {{\pi \over 4} - 2x} }} = {{2\sin \sqrt {\pi - 8x} } \over {\sqrt {\pi - 8x} }} \cr} \)
f. \(y’ = \sqrt {\sin 3x} + x.{{3\cos 3x} \over {2\sqrt {\sin 3x} }} = {{2\sin 3x + 3x\cos 3x} \over {2\sqrt {\sin 3x} }}\)