第241题

A.\({y}^{，}=2x\cos {2x}-{x}^{2}\sin {2x}\)

B.\({y}^{，}={x}^{2}\cos {2x}-2x\sin {2x}\)

C.\({y}^{，}=2x\cos {2x}-2{x}^{2}\sin {2x}\)

D.\({y}^{，}=2x\cos {2x}+2{x}^{2}\sin {2x}\)

参考答案：C

第242题

A.\({y}^{，}=-\frac {3} {{x}^{2}}\sin^{2} {\frac {1} {x}}\)

B.\({y}^{，}=-\frac {3} {{2x}^{2}}\sin^{2} {\frac {2} {x}}\)

C.\({y}^{，}=-\frac {3} {{x}^{2}}\cos {\frac {1} {x}}\sin^{2} {\frac {1} {x}}\)

D.\({y}^{，}=-\frac {3} {{2x}^{2}}\sin {\frac {1} {x}\sin {\frac {2} {x}}}\)

参考答案：A

第243题

A.\(\left [ {\ln {\left ( {2x+1} \right )}} \right ]^{\, '}=\frac {2} {2x+1}\)

B.\(\left ( {{e}^{5x-4}} \right )^{\, '}={e}^{5x-4}\)

C.\(\left ( {\sqrt {2x-1}} \right )^{\, '}=\frac {1} {\sqrt {2x-1}}\)

D.\(\left [ {\cos {\left ( {2x+\frac {\pi } {3}} \right )}} \right ]^{\, '}=2\sin {\left ( {2x+\frac {\pi } {3}} \right )}\)

参考答案：AC

第244题

参考答案：\(-\frac {x+2} {\left ( {2x-1} \right )^{2}\sqrt {1+{x}^{2}}}\)

第245题

参考答案：\(-{2}^{-x}{f}^{\, '}\left ( {{2}^{-x}} \right )\ln {2}\)

第246题

A.若\(y=\cos {\frac {1} {x}}\)，则\({y}^{，}=-\frac {1} {{x}^{2}}\sin {\frac {1} {x}}\)

B.若\(y=\sin {{x}^{2}}\)，则\({y}^{，}=2x\cos {{x}^{2}}\)

C.若\(y=\cos {5x}\)，则\({y}^{，}=-5\sin {5x}\)

D.若\(y=\frac {1} {2}x\sin {2x}\)，则\({y}^{，}=x\sin {2x}\)

参考答案：BC

第247题

参考答案：10

解析：

【详解】

\({f}^{\, '}\left ( {x} \right )=\left ( {\left ( {x+\sqrt {1+{x}^{2}}} \right )^{10}} \right )^{\, '}=10\left ( {x+\sqrt {1+{x}^{2}}} \right )^{9}\left ( {1+\frac {1} {2}\cdot \frac {2x} {\sqrt {1+2{x}^{2}}}} \right )\)

\(\therefore {f}^{\, '}\left ( {0} \right )=10，f\left ( {0} \right )=1⇒\frac {{f}^{\, '}\left ( {0} \right )} {f\left ( {0} \right )}=10\)

第248题

参考答案：\({y}^{，}=2\sin {\left ( {4x+\frac {2\pi } {3}} \right )}\)

解析：\({y}^{，}=2\sin {\left ( {2x+\frac {\pi } {3}} \right )}\cdot 2\cos {\left ( {2x+\frac {\pi } {3}} \right )}=2\sin {\left ( {4x+\frac {2\pi } {3}} \right )}\)

第249题

参考答案：\({y}^{，}=-2\sin {4x}\)

解析：

第250题

参考答案：\({y}^{，}=\frac {2x} {\left ( {1-2{x}^{2}} \right )\sqrt {1-2{x}^{2}}}\)

解析：

第251题

A.\(\frac {1-\ln {x}} {{x}^{2}}\)

B.\(\frac {1+\ln {x}} {{x}^{2}}\)

C.\(\frac {\ln {x+1}} {x}\)

D.\(\frac {\ln {x-1}} {x}\)

参考答案：A

第252题

A.\({y}^{，}=2x\cos {x}-{x}^{2}\sin {x}\)

B.\({y}^{，}=-2x\sin {x}\)

C.\({y}^{，}=2x\cos {x}+{x}^{2}\sin {x}\)

D.\({y}^{，}=x\cos {x}-{x}^{2}\sin {x}\)

参考答案：A

第253题

A.\(\left ( {{x}^{2}+{2}^{x}} \right )^{'}=2x+{2}^{x}\ln {2}\)

B.\(\left ( {{x}^{2}{e}^{x}} \right )^{'}=\left ( {2x+{x}^{2}} \right ){e}^{x}\)

C.\(\left ( {\frac {{x}^{2}} {\ln {x}}} \right )^{'}=\frac {x-2\ln {x}} {\text{ln}^{2}x}\)

D.\(\left ( {{x}^{3}-\frac {1} {x}} \right )^{'}=3{x}^{2}+\frac {1} {{x}^{2}}\)

参考答案：ABD

第254题

A.\(\left ( {x\cos {x}} \right )^{'}=\cos {x-x\sin {x}}\)

B.\(\left ( {x\ln {x}} \right )^{'}=\ln {x}+1\)

C.\(\left ( {{e}^{x}\sin {x}} \right )^{'}={e}^{x}\left ( {\sin {x+\cos {x}}} \right )\)

D.\(\left ( {\frac {\cos {x}} {{e}^{x}}} \right )^{'}=\frac {\sin {x-\cos {x}}} {{e}^{x}}\)

参考答案：ABC

第255题

参考答案：\(\frac {1} {\cos^{2} {x}}\)

第256题

参考答案：设\(g\left ( {x} \right )=\left ( {x+1} \right )\left ( {x+2} \right )\left ( {x+3} \right )\left ( {x+4} \right )\left ( {x+5} \right )\)，则\(f\left ( {x} \right )=xg\left ( {x} \right )\)

所以\({f}^{\, '}\left ( {x} \right )=g\left ( {x} \right )+x{g}^{\, '}\left ( {x} \right )\)，所以\({f}^{\, '}\left ( {0} \right )=g\left ( {0} \right )=1\times 2\times 3\times 4\times 5=120\).

第257题

参考答案：设切点坐标为\(\left ( {{x}_{0}，{x}^{3}_{0}-3{x}^{2}_{0}+2} \right )\)，\({y}^{，}=3{x}^{2}-6x+2\)，则\(\left \{ \begin{gathered} {k=3{x}^{2}_{0}-6{x}_{0}+2} \\ {{k{x}_{0}=x}^{3}_{0}-3{x}^{2}_{0}+2{x}_{0}} \end{gathered} \right .\)

解得\(k=2\)或\(-\frac {1} {4}\).

第258题

参考答案：\(\frac {2} {3}\)

第259题

A.\( (\mathrm{cos}\frac{\pi }{3}{)}^{\text{'}}=-\mathrm{sin}\frac{\pi }{3}\)

B.\( ({2}^{x}{)}^{\mathrm{\text{'}}}=x\cdot {2}^{x-1}\)

C.\( (\mathrm{sin}x{)}^{\mathrm{\text{'}}}=-\mathrm{cos}x\)

D.\( (\sqrt{x}{)}^{\text{'}}=\frac{1}{2\sqrt{x}}\)

参考答案：D

第260题

A.\( (\mathrm{ln}2{)}^{\text{'}}=\frac{1}{2}\)

B.\( (\frac{1}{{x}^{2}}{)}^{\text{'}}=\frac{2}{{x}^{3}}\)

C.\( (\mathrm{cos}x{)}^{\text{'}}=-\mathrm{sin}x\)

D.\( ({3}^{x}{)}^{\text{'}}={3}^{x}{\mathrm{log}}_{3}e\)

参考答案：ABD