数学の基礎と公式-4

数学の基礎と公式

amazon kindle版を出版しました。

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4. 微分

(1)$(x)'=1,~(x^2)'=2x,~(x^3)'=3x^2,\cdots,~(x^n)'=nx^{n-1}$

(2)$(\sqrt{x})'={\dfrac{1}{2\,\sqrt{x}}},~(\dfrac{1}{\sqrt{x}})'=-{\dfrac{1}{2\,x^{{{3}\over{2}}}}},~(\dfrac{1}{\sqrt{x}})''={\dfrac{3}{4\,x^{{{5}\over{2}}}}} $

(3)$(\log x)'=\dfrac{1}{x},~(\log_y x)'=\left(\dfrac{\log x}{\log y} \right)'=\dfrac{1}{x \log y}$

(4)$(e^x)'=e^x,~(e^{x^2})'=2x e^{x^2},~(e^{x^x})'=x^{x}\,e^{x^{x}}\,\left(\log x+1\right)$

(5) $(y^x)'=y^{x}\,\log y$

(6)$(\sin x)'=\cos x,~(\cos x)'=-\sin x,~(\tan x)'=\dfrac{1}{\cos^2 x}$

(7)$(\mathrm{cosec}\, x)'=-\cot x \,\mathrm{cosec}\, x,\, ~(\sec x)'=\sec x\,\tan x,\,~(\cot x)'=-\left(\mathrm{cosec}\, x \right)^2$

(8)$(\sin^{-1} x)'={\dfrac{1}{\sqrt{1-x^2}}},~(\cos^{-1} x)'=-{\dfrac{1}{\sqrt{1-x^2}}},~(\tan^{-1} x)'={\dfrac{1}{x^2+1}}$

(9)$(\mathrm{cosec}^{-1}\, x)'=-{\dfrac{1}{\sqrt{1-{\dfrac{1}{x^2}}}\,x^2}}$

$(\sec^{-1}\, x)'={\dfrac{1}{\sqrt{1-{\dfrac{1}{x^2}}}\,x^2}}$

$(\cot^{-1}\, x)'=-{\dfrac{1}{x^2+1}}$

(10) $(\mathrm{sin h} x )'=\mathrm{cos h} x $ ,

$(\mathrm{cos h}(x))'=\mathrm{sin h} x $ ,

$(\mathrm{tan h}(x))'=(\mathrm{sec h}(x))^2$

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n次導関数

(1)$(x^n)^{(m)}=n(n-1)(n-2) \cdot (n-m+1)x^{(n-m)}$

$(x^n)^{(n)}=n!$

(2)$(e^x)^{n}=e^x$

(3)$(y^x)^{n}=y^x(\log y)^n$

(4)$(\log x)^{n}=\dfrac{(-1)^{n-1}(n-1)!}{x^n}$

(5)

$(\sin x)^{n}=\sin \left(\dfrac{n \pi}{2}+x \right)$

$(\cos x)^{n}=\cos \left(\dfrac{n \pi}{2}+x \right)$

(6)$(e^x \sin x)^{n}=\left( \sin \dfrac{\pi}{4}\right)^{-n} e^x \sin \left(\dfrac{n \pi}{4}+x \right)$

(7)$(e^x \cos x)^{n}=\left( \sin \dfrac{\pi}{4}\right)^{-n} e^x \cos \left(\dfrac{n \pi}{4}+x \right)$