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Bài 4. Tìm đạo hàm của các hàm số sau:
a) \(y = \left( {9 – 2x} \right)(2{x^3} – 9{x^2} + 1)\);
b) \(y = \left ( 6\sqrt{x} -\frac{1}{x^{2}}\right )(7x -3)\);
c) \(y = (x -2)\sqrt{(x^2+1)}\);
d) \(y = tan^2x +cotx^2\);
e) \(y = cos\frac{x}{1+x}\).
a) \(y’ = \left( {9 – 2x} \right)'(2{x^3} – 9{x^2} + 1) + \left( {9 – 2x} \right)(2{x^3} – 9{x^2} + 1)’\)
\(= – 2(2{x^3} – 9{x^2} + 1) + \left( {9 – 2x} \right)(6{x^2} – 18x) \)
\(= – 16{x^3} + 108{x^2} – 162x – 2\).
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b) \(y’ = \left ( 6\sqrt{x} -\frac{1}{x^{2}}\right )’.(7x -3) +\left ( 6\sqrt{x} -\frac{1}{x^{2}}\right )(7x -3)’\)
\(= \left ( \frac{3}{\sqrt{x}} +\frac{2}{x^{3}}\right )(7x -3) +7 \left ( 6\sqrt{x} -\frac{1}{x^{2}}\right )\).
c) \(y’ = (x -2)’\sqrt{(x^2+1)} + (x -2)\sqrt {(x^2+1)}’ \)
\(= \sqrt {(x^2+1)} + (x -2)\frac{\left ( x^{2}+1 \right )’}{2\sqrt{x^{2}+1}}\)
\(= \sqrt {(x^2+1)} + (x -2) \frac{2x}{2\sqrt{x^{2}+1}}\)
\( = \sqrt {(x^2+1)} + \frac{x^{2}-2x}{\sqrt{x^{2}+1}}\) = \( \frac{2x^{2}-2x+1}{\sqrt{x^{2}+1}}\).
d) \(y’ = 2tanx.(tanx)’ – (x^2)’ \left ( -\frac{1}{sin^{2}x^{2}} \right )\) = \( \frac{2tanx}{cos^{2}x}+\frac{2x}{sin^{2}x^{2}}\).
e) \(y’ = \left ( \frac{1}{1+x} \right )’sin \frac{x}{1+x}\) = \( -\frac{1}{(1+x)^{2}}sin \frac{x}{1+x}\).