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导数公式 #18

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zhouhaibing089 opened this issue Nov 6, 2016 · 1 comment
Open

导数公式 #18

zhouhaibing089 opened this issue Nov 6, 2016 · 1 comment
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@zhouhaibing089
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zhouhaibing089 commented Nov 6, 2016 •
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常数函数

f(x)=C
f'(x)=\lim_{\Delta x\to0}{{f(x_0+\Delta x)-f(x_0)}\over\Delta x}={{C-C}\over\Delta x}=0

代数函数

f(x)=x^n
f'(x)=\lim_{\Delta x\to0}{{(x_0+\Delta x)^n-x_0^n}\over \Delta x}
f'(x)=\lim_{\Delta x \to0}{{{n \choose 0}x_0^n\Delta x^0+{n \choose 1}x_0^{n-1}\Delta x^1+\cdots+{n \choose {n-1}}x_0^1\Delta x^{n-1}+{n \choose n}x_0^0\Delta x^n-x_0^n}\over \Delta x}
f'(x)=\lim_{\Delta x \to0}{{n \choose 1}x_0^{n-1}\Delta x^0+{{n \choose 2}x_0^{n-2}\Delta x^1}+\cdots+{n \choose {n-1}}x_0^1\Delta x^{n-2}+{n \choose n}x_0^0\Delta x^{n-1}}
f'(x)={n \choose 1}x_0^{n-1}=nx_0^{n-1}

对数函数

f(x)=\ln x
f'(x)={\lim_{\Delta x\to0}}{{\ln(x_0+\Delta x)-\ln(x_0)}\over\Delta x}=\ln(1+{\Delta x\over x_0})^{1\over \Delta x}
令:
u={x_0\over\Delta x}\Rightarrow u\to \infty
f'(x)=\ln(1+{1\over u})^{u{1\over x_0}}={1\over x_0}\ln(1+{1\over u})^u
所以:
f'(x)={1\over x_0}\ln e={1\over x_0}

指数函数

f(x)=e^x
y=\ln e^x=x \Rightarrow y'={1\over e^x}(e^x)'=x'=1 \Rightarrow f'(x)=(e^x)'=e^x

正弦函数

f(x)=\sin x
f'(x)=\lim_{\Delta x\to0}{{\sin(x_0+\Delta x)-\sin x_0}\over\Delta x}
f'(x)=\lim_{\Delta x\to0}{{\sin x_0\cos\Delta x+\cos x_0\sin\Delta x-\sin x_0}\over\Delta x}
f'(x)=\lim_{\Delta x\to0}{{\cos x_0\sin\Delta x}\over\Delta x}=\cos x_0

余弦函数

f(x)=\cos x
f'(x)=\lim_{\Delta x\to0}{{\cos(x_0+\Delta x)-\cos x_0}\over\Delta x}
f'(x)=\lim_{\Delta x\to0}{{\cos x_0\cos\Delta x-\sin x_0\sin\Delta x-\cos x_0}\over\Delta x}
f'(x)=\lim_{\Delta x\to0}{{-\sin x_0\sin\Delta x}\over\Delta x}=-\sin x_0

正切函数

f(x)=\tan x={\sin x \over \cos x}
f'(x)={{\sin'x\cos x-\sin x\cos'x}\over{\cos^2x}}={{\cos^2x+\sin^2x}\over\cos^2x}
f'(x)={1\over\cos^2x}={\sec^2x}

余切函数

f(x)=\cot x={\cos x\over \sin x}
f'(x)={{\cos'x\sin x-\cos x\sin'x}\over\sin^2x}={{-\sin^2x-\cos^2x}\over\sin^2x}
f'(x)={-1\over\sin^2x}=-\csc^2x

正割函数

f(x)=\sec x={1\over\cos x}
f'(x)=-\cos^{-2}x\cos'x={\sin x\over\cos^2x}=\tan x\sec x

余割函数

f(x)=\csc x={1\over\sin x}
f'(x)=-sin^{-2}x\sin'x={-\cos x\over\sin^2x}=-\cot x\csc x

反正弦函数

f(x)=\arcsin x,f(x)\in({-\pi\over2},{\pi\over2})
y=\sin(\arcsin x)=x\Rightarrow y'=\cos(\arcsin x)\arcsin'x=1
f'(x)={1\over\cos(\arcsin x)}={1\over\sqrt{1-\sin^2{\arcsin x}}}={1\over\sqrt{1-x^2}}

反余弦函数

f(x)=\arccos x={\pi\over2}-\arcsin x,f(x)\in(0,\pi)
f'(x)=-\arcsin'x=-{1\over\sqrt{1-x^2}}

反正切函数

f(x)=\arctan x,f(x)\in({-\pi\over2},{\pi\over2})
y=\tan(\arctan x)=x\Rightarrow y'=\sec^2(\arctan x)\arctan'x=1
f'(x)={1\over\sec^2(\arctan x)}={1\over{1+\tan^2(\arctan x)}}={1\over{1+x^2}}

反余切函数

f(x)=\operatorname{arccot} x={\pi\over2}-\arctan x,f(x)\in(0,\pi)
f'(x)=-\arctan'x=-{1\over{1+x^2}}

反正割函数

f(x)=\operatorname{arcsec}x,f(x)\in[0,{\pi\over2})\cup({\pi\over2},\pi]
y=\sec(\operatorname{arcsec}x)=x\Rightarrow y'=\tan(\operatorname{arcsec}x)\sec(\operatorname{arcsec}x)\operatorname{arcsec}'x=1
f'(x)={1\over{|x|\tan(\operatorname{arcsec}x)}}={1\over{|x|\sqrt{\sec^2(\operatorname{arcsec}x)-1}}}={1\over{|x|\sqrt{x^2-1}}}

反余割函数

f(x)=\operatorname{arccsc} x={\pi\over2}-\operatorname{arcsec}x,f(x)\in[-{\pi\over2},0)\cup(0,{\pi\over2}]
f'(x)=-\operatorname{arcsec}'x=-{1\over{|x|\sqrt{x^2-1}}}

参考资料
@zhouhaibing089 zhouhaibing089 added blog and removed blog labels Dec 11, 2016
@xiaoyu2er
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海兵哥哥太帅了

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