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Câu hỏi: Tìm đạo hàm của các hàm số sau:
\(\begin{array}{l}
a) y = \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)\\
b) y = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)\\
c) y = \left({x - 2} \right)\sqrt {{x^2} + 1} \\
d) y = {\tan ^2}x - {\cot}{x^2}\\
e) y = \cos \dfrac{x}{{1 + x}}
\end{array}\)
\(\begin{array}{l}
a) y = \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)\\
b) y = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)\\
c) y = \left({x - 2} \right)\sqrt {{x^2} + 1} \\
d) y = {\tan ^2}x - {\cot}{x^2}\\
e) y = \cos \dfrac{x}{{1 + x}}
\end{array}\)
Phương pháp giải
Sử dụng các quy tắc tính đạo hàm của tích, thương, quy tắc tính đạo hàm hàm số hợp và bảng đạo hàm cơ bản.
Lời giải chi tiết
\(\begin{array}{l}
a) y = \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)\\y' = \left({9 - 2x} \right)'\left({2{x^3} - 9{x^2} + 1} \right) \\+ \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)'\\
= - 2\left({2{x^3} - 9{x^2} + 1} \right) + \left({9 - 2x} \right)\left({6{x^2} - 18x} \right)\\
= - 4{x^3} + 18{x^2} - 2 + 54{x^2} - 162x - 12{x^3} + 36{x^2}\\
= - 16{x^3} + 108{x^2} - 162x - 2\\
b) y = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)\\y' = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)'\left({7x - 3} \right) + \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)'\\
= \left({6.\dfrac{1}{{2\sqrt x }} - \dfrac{{ - \left( {{x^2}} \right)'}}{{{{\left({{x^2}} \right)}^2}}}} \right)\left({7x - 3} \right) + \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right). 7\\ = \left({\dfrac{3}{{\sqrt x }} + \dfrac{{2x}}{{{x^4}}}} \right)\left({7x - 3} \right) + 7\left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\\= \left({\dfrac{3}{{\sqrt x }} + \dfrac{2}{{{x^3}}}} \right)\left({7x - 3} \right) + 7\left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\\
= 21\sqrt x - \dfrac{9}{{\sqrt x }} + \dfrac{{14}}{{{x^2}}} - \dfrac{6}{{{x^3}}} + 42\sqrt x - \dfrac{7}{{{x^2}}}\\
= \dfrac{{ - 6}}{{{x^3}}} + \dfrac{7}{{{x^2}}} + 63\sqrt x - \dfrac{9}{{\sqrt x }}\\
c) y = \left({x - 2} \right)\sqrt {{x^2} + 1} \\y' = \left({x - 2} \right)'\sqrt {{x^2} + 1} + \left({x - 2} \right)\left({\sqrt {{x^2} + 1} } \right)'\\ = 1.\sqrt {{x^2} + 1} + \left({x - 2} \right).\dfrac{{\left({{x^2} + 1} \right)'}}{{2\sqrt {{x^2} + 1} }} \\= \sqrt {{x^2} + 1} + \left({x - 2} \right).\dfrac{{2x}}{{2\sqrt {{x^2} + 1} }}\\
= \sqrt {{x^2} + 1} + \left({x - 2} \right)\dfrac{x}{{\sqrt {{x^2} + 1} }}\\
= \dfrac{{{x^2} + 1 + {x^2} - 2x}}{{\sqrt {{x^2} + 1} }}\\
= \dfrac{{2{x^2} - 2x + 1}}{{\sqrt {{x^2} + 1} }}\\
d) y = {\tan ^2}x - \cot {x^2}\\y' = \left({{{\tan }^2}x} \right)' - \left({\cot {x^2}} \right)'\\ = 2\tan x.\left({\tan x} \right)' - \left({{x^2}} \right)'.\dfrac{{ - 1}}{{\sin ^2 {x^2}}}\\
= 2\tan x.\dfrac{1}{{{{\cos }^2}x}} + \dfrac{{2x}}{{{{\sin }^2}x^2}}\\
= \dfrac{{2\sin x}}{{{{\cos }^3}x}} + \dfrac{{2x}}{{{{\sin }^2}x^2}}\\
e)y = \cos \dfrac{x}{{1 + x}}\\y' = \left({\dfrac{x}{{x + 1}}} \right)'.\left({ - \sin \dfrac{x}{{x + 1}}} \right)\\ = - \sin \left({\dfrac{x}{{1 + x}}} \right).\dfrac{{\left(x \right)'\left({1 + x} \right) - x.\left({1 + x} \right)'}}{{{{\left({1 + x} \right)}^2}}}\\
= - \sin \dfrac{x}{{1 + x}}.\left({\dfrac{{1 + x - x}}{{{{\left( {1 + x} \right)}^2}}}} \right)\\
= - \dfrac{1}{{{{\left({1 + x} \right)}^2}}}.\sin \dfrac{x}{{1 + x}}
\end{array}\)
Sử dụng các quy tắc tính đạo hàm của tích, thương, quy tắc tính đạo hàm hàm số hợp và bảng đạo hàm cơ bản.
Lời giải chi tiết
\(\begin{array}{l}
a) y = \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)\\y' = \left({9 - 2x} \right)'\left({2{x^3} - 9{x^2} + 1} \right) \\+ \left({9 - 2x} \right)\left({2{x^3} - 9{x^2} + 1} \right)'\\
= - 2\left({2{x^3} - 9{x^2} + 1} \right) + \left({9 - 2x} \right)\left({6{x^2} - 18x} \right)\\
= - 4{x^3} + 18{x^2} - 2 + 54{x^2} - 162x - 12{x^3} + 36{x^2}\\
= - 16{x^3} + 108{x^2} - 162x - 2\\
b) y = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)\\y' = \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)'\left({7x - 3} \right) + \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\left({7x - 3} \right)'\\
= \left({6.\dfrac{1}{{2\sqrt x }} - \dfrac{{ - \left( {{x^2}} \right)'}}{{{{\left({{x^2}} \right)}^2}}}} \right)\left({7x - 3} \right) + \left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right). 7\\ = \left({\dfrac{3}{{\sqrt x }} + \dfrac{{2x}}{{{x^4}}}} \right)\left({7x - 3} \right) + 7\left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\\= \left({\dfrac{3}{{\sqrt x }} + \dfrac{2}{{{x^3}}}} \right)\left({7x - 3} \right) + 7\left({6\sqrt x - \dfrac{1}{{{x^2}}}} \right)\\
= 21\sqrt x - \dfrac{9}{{\sqrt x }} + \dfrac{{14}}{{{x^2}}} - \dfrac{6}{{{x^3}}} + 42\sqrt x - \dfrac{7}{{{x^2}}}\\
= \dfrac{{ - 6}}{{{x^3}}} + \dfrac{7}{{{x^2}}} + 63\sqrt x - \dfrac{9}{{\sqrt x }}\\
c) y = \left({x - 2} \right)\sqrt {{x^2} + 1} \\y' = \left({x - 2} \right)'\sqrt {{x^2} + 1} + \left({x - 2} \right)\left({\sqrt {{x^2} + 1} } \right)'\\ = 1.\sqrt {{x^2} + 1} + \left({x - 2} \right).\dfrac{{\left({{x^2} + 1} \right)'}}{{2\sqrt {{x^2} + 1} }} \\= \sqrt {{x^2} + 1} + \left({x - 2} \right).\dfrac{{2x}}{{2\sqrt {{x^2} + 1} }}\\
= \sqrt {{x^2} + 1} + \left({x - 2} \right)\dfrac{x}{{\sqrt {{x^2} + 1} }}\\
= \dfrac{{{x^2} + 1 + {x^2} - 2x}}{{\sqrt {{x^2} + 1} }}\\
= \dfrac{{2{x^2} - 2x + 1}}{{\sqrt {{x^2} + 1} }}\\
d) y = {\tan ^2}x - \cot {x^2}\\y' = \left({{{\tan }^2}x} \right)' - \left({\cot {x^2}} \right)'\\ = 2\tan x.\left({\tan x} \right)' - \left({{x^2}} \right)'.\dfrac{{ - 1}}{{\sin ^2 {x^2}}}\\
= 2\tan x.\dfrac{1}{{{{\cos }^2}x}} + \dfrac{{2x}}{{{{\sin }^2}x^2}}\\
= \dfrac{{2\sin x}}{{{{\cos }^3}x}} + \dfrac{{2x}}{{{{\sin }^2}x^2}}\\
e)y = \cos \dfrac{x}{{1 + x}}\\y' = \left({\dfrac{x}{{x + 1}}} \right)'.\left({ - \sin \dfrac{x}{{x + 1}}} \right)\\ = - \sin \left({\dfrac{x}{{1 + x}}} \right).\dfrac{{\left(x \right)'\left({1 + x} \right) - x.\left({1 + x} \right)'}}{{{{\left({1 + x} \right)}^2}}}\\
= - \sin \dfrac{x}{{1 + x}}.\left({\dfrac{{1 + x - x}}{{{{\left( {1 + x} \right)}^2}}}} \right)\\
= - \dfrac{1}{{{{\left({1 + x} \right)}^2}}}.\sin \dfrac{x}{{1 + x}}
\end{array}\)