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| （未顯示由 15 位使用者於中間所作的 197 次修訂） |
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| {{SandboxNote}}
| | #REDIRECT [[主命名空间沙盒]] |
| <math>
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| 24.
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| (1)x^{2}+y^{2}-xy=1
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| 解：2x+2y\cdot y^{'}=y+x\cdot y^{'}
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| y^{'}\left ( 2y-x\right )=y-2x
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| y=\frac{y-2x}{2y-x}
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| (4)y=1+xe^{y}
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| 解：y^{'}=e^{y}+e^{y}\cdot y^{'}\cdot x
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| y^{'}\left ( 1-e^{y}\cdot x\right )=e^{y}
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| y^{'}=\frac{e^{y}}{1-e^{y}\cdot x}
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| 27.
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| (1)y=x\sqrt{\frac{1-x}{1+x}}
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| 解：\ln y=\ln x+\frac{1}{2}\ln \left | 1-x\right |-\frac{1}{2}\left | 1+x\right |
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| \frac{y^{'}}{y}=\frac{1}{x}-\frac{1}{2}\cdot \frac{1}{1-x}-\frac{1}{2}\cdot \frac{1}{1+x}=\frac{1}{x}-\frac{1}{1-x^{2}}
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| y^{'}=x\sqrt{\frac{1-x}{1+x}}\cdot \left ( \frac{1}{x}-\frac{1}{1-x^{2}}\right )
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| =\frac{x\cdot \sqrt{1-x^{2}}}{1+x}\cdot \left ( \frac{1}{x}-\frac{1}{1-x^{2}}\right )
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| =\frac{\sqrt{1-x^{2}}}{1+x}-\frac{x\sqrt{1-x^{2}}}{\left ( 1+x\right )\left ( 1-x^{2}\right )}
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| =\frac{\sqrt{1-x^{2}}}{1+x}\left ( 1-\frac{1}{1-x^{2}}\right )
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| =-\frac{x^{2}\sqrt{1-x^{2}}}{\left ( 1+x\right )\left ( 1-x^{2}\right )}
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| (5)y=\left ( \sin x\right )^{\tan x}
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| 解：\ln y=\tan x\ln \sin x
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| \frac{y^{'}}{y}=\frac{\ln \sin x}{\sin ^{2}x}+\frac{\cos x}{\sin x}\cdot \tan x=\frac{\ln \sin x}{\sin ^{2}x}+1
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| y^{'}=\left ( \sin x\right )^{\tan x}\cdot \left ( \frac{\ln \sin x}{\sin ^{2}x}+1\right )
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| =\left ( \sin x\right )^{\tan x-2}\ln \sin x+\left ( \sin x\right )^{\tan x}
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| 28.
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| 解：y^{\sin x}=\left ( \sin x\right )^{y}
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| \ln y^{\sin x}=\ln \left ( \sin x\right )^{y}
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| \sin x\ln y=y\ln \sin x
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| \cos x\ln y+\left ( \ln y\right )^{'}\sin x=y^{'}\ln \sin x+\left ( \ln \sin x\right )^{'}y
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| \cos x\ln y+y^{'}\frac{\sin x}{y}=y^{'}\ln \sin x+\frac{y\cdot \cos x}{\sin x}
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| y^{'}\left ( \frac{\sin x}{y}-\ln \sin x\right )=\frac{y\cdot \cos x}{\sin x}-\cos x\ln y
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| y^{'}\left ( \frac{\sin x-y\ln \sin x}{y}\right )=\frac{y-\sin x \ln y}{\tan x}
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| y^{'}=\frac{y\left ( y-\sin x\ln y\right )}{\tan x\left ( \sin x-y\ln \sin x\right )}
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| 57.
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| (3)y=\ln x^{2}
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| 解：\mathrm{d}y=\mathrm{d}\left ( \ln x^{2}\right )
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| =\frac{1}{x^{2}}\mathrm{d}x^{2}
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| =\frac{2x}{x^{2}}\mathrm{d}x=\frac{2}{x}\mathrm{d}x
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| (5)y=e^{-x}\cos x
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| 解：\mathrm{d}y=\mathrm{d}\left ( e^{-x}\cos x\right )
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| \mathrm{d}y=\cos x\mathrm{d}e^{-x}+e^{-x}\mathrm{d}\cos x
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| \mathrm{d}y=-e^{-x}\cos x\mathrm{d}x+e^{-x}\sin x\mathrm{d}x
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| \mathrm{d}y=e^{-x}\left ( \sin x-\cos x\right )\mathrm{d}x
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| (8)y=\left ( e^{x}+e^{-x}\right )^{2}
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| 解：\ln y=2\ln \left ( e^{x}+e^{-x}\right )
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| \frac{\mathrm{d}y}{y}=\frac{2}{e^{x}+e^{-x}}\mathrm{d}\left ( e^{x}+e^{-x}\right )
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| =\frac{2}{e^{x}+e^{-x}}\left ( \mathrm{d}e^{x}+\mathrm{d}e^{-x}\right )
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| =\frac{2}{e^{x}+e^{-x}}\left ( e^{x}\mathrm{d}x-e^{-x}\mathrm{d}x\right )
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| =\frac{2\left ( e^{x}-e^{-x}\right )}{e^{x}+e^{-x}}\mathrm{d}x
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| \mathrm{d}y=\frac{2\left ( e^{x}-e^{-x}\right )\left ( e^{x}+e^{-x}\right )^{2}}{e^{x}+e^{-x}}\mathrm{d}x
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| =2\left ( e^{2x}-e^{-2x}\right )\mathrm{d}x
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| 58.
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| 解：xy=e^{x+y}
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| y=\frac{e^{x+y}}{x}
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| \ln y=x+y-\ln x
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| \frac{\mathrm{d}y}{y}=\mathrm{d}x+\mathrm{d}y+\frac{1}{x}\mathrm{d}x
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| \mathrm{d}y\left ( \frac{y-1}{y}\right )=\frac{x+1}{x}\mathrm{d}x=\frac{y\left ( x+1\right )}{x\left ( y-1\right )}\mathrm{d}x
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| 62.
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| (5)解：\cos 60^{\circ}20^{'}
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| \approx \cos 60^{\circ}-\sin 60^{\circ}\cdot \frac{20}{180\times 360}\pi =0.49916
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| </math>
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| ----
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| <math>
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| 7.
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| 解：要证nb^{n-1}\left ( a-b\right )< a^{n}-b^{n}< na^{n-1}\left ( a-b\right )
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| 因为a> b
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| 所以只要证nb^{n-1}<a^{n-1}+ba^{n-2}+b^{2}a^{n-3}…+b^{n-1}<na^{n-1}
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| 因为a>b
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| 所以a^{n-1}+ba^{n-2}+b^{2}a^{n-3}…+b^{n-1}<a^{n-1}+aa^{n-2}+a^{2}a^{n-3}…+a^{n-1}=na^{n-1}
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| 同理a^{n-1}+ba^{n-2}+b^{2}a^{n-3}…+b^{n-1}>nb^{n-1}
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| 故得证
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| 9.
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| (1)\lim_{x\rightarrow 0}\frac{e^{x}-e^{-x}}{x}
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| 解：原式=\lim_{x\rightarrow 0}\frac{e^{x}+e^{-x}}{1}=2
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| (2)\lim_{x\rightarrow 1}\frac{\ln x}{x-1}
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| 解：原式=\lim_{x\rightarrow 1}\frac{\frac{1}{x}}{1}=1
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| (3)\lim_{x\rightarrow 1}\frac{x^{3}-3x^{2}+2}{x^{3}-x^{2}-x+1}
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| 解：原式=\lim_{x\rightarrow 1}\frac{3x^{2}-6x}{3x^{2}-2x-1}=\infty
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| (8)\lim_{x\rightarrow 0^{+}}x^{m}\ln x\left ( m>0\right )
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| 解：原式=\lim_{x\rightarrow 0^{+}}\frac{\ln x}{x^{-m}}
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| =\lim_{x\rightarrow 0^{+}}\frac{\frac{1}{x}}{-mx^{-m-1}}
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| =\lim_{x\rightarrow 0^{+}}-mx^{m+2}=0
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| (9)\lim_{x\rightarrow 0}\left ( \frac{1}{x}-\frac{1}{e^{x}-1}\right )
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| 解：原式=\lim_{x\rightarrow 0}\frac{e^{x}-1-x}{xe^{x}-x}
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| =\lim_{x\rightarrow 0}\frac{e^{x}-1}{e^{x}+xe^{x}-1}
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| =\lim_{x\rightarrow 0}\frac{e^{x}}{2e^{x}+xe^{x}}=\frac{1}{2}
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| (10)\lim_{x\rightarrow 0}\left ( 1+\sin x\right )^{\frac{1}{x}}
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| 解：原式=\lim_{x\rightarrow 0}e^{\ln \left ( 1+\sin x\right )^{\frac{1}{x}}}
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| =\lim_{x\rightarrow 0}e^{\frac{\ln \left ( 1+\sin x\right )}{x}}
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| =e^{\lim_{x\rightarrow 0}\frac{\ln \left ( 1+\sin x\right )}{x}}
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| =e^{\lim_{x\rightarrow 0}\frac{\cos x}{1+\sin x}}
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| =e^{\frac{1}{1+0}}=e
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| (12)\lim_{x\rightarrow 0^{+}}x^{\sin x}
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| 解：原式=\lim_{x\rightarrow 0^{+}}e^{\ln x^{\sin x}}
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| =\lim_{x\rightarrow 0^{+}}e^{\frac{\ln x}{\frac{1}{\sin x}}}
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| =e^{\lim_{x\rightarrow 0^{+}}\frac{\ln x}{\frac{1}{\sin x}}}
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| =e^{\lim_{x\rightarrow 0^{+}}\frac{\ln x}{\frac{1}{x}}}
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| =e^{\lim_{x\rightarrow 0^{+}}\frac{x^{-1}}{-x^{-2}}}
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| =e^{\lim_{x\rightarrow 0^{+}}-x}
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| =e^{-0}
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| =1
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| </math>
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