三角関数と双曲線関数の微分
三角関数の微分
(1)
\[ \frac{d}{dx}\sin x=\cos x \](2)
\[ \frac{d}{dx}\cos x=-\sin x \](3)
\[ \frac{d}{dx}\tan x=\cos^{-2}x \](4)
\[ \frac{d}{dx}\sin^{-1}x=-\sin^{-1}x\tan^{-1}x \](5)
\[ \frac{d}{dx}\cos^{-1}x=\cos^{-1}x\tan x \](6)
\[ \frac{d}{dx}\tan^{-1}x=-\sin^{-2}x \](1)
\begin{align*} \frac{d}{dx}\sin x & =\frac{d}{dx}\frac{e^{ix}-e^{-ix}}{2i}\\ & =\frac{ie^{ix}+ie^{-ix}}{2i}\\ & =\cos x \end{align*}(2)
\begin{align*} \frac{d}{dx}\cos x & =\frac{d}{dx}\frac{e^{ix}+e^{-ix}}{2}\\ & =\frac{ie^{ix}-ie^{-ix}}{2}\\ & =-\frac{e^{ix}-e^{-ix}}{2i}\\ & =-\sin x \end{align*}(3)
\begin{align*} \frac{d}{dx}\tan x & =\frac{d}{dx}\frac{\sin x}{\cos x}\\ & =\frac{(\sin x)'\cos x-\sin x(\cos x)'}{\cos^{2}x}\\ & =\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}\\ & =\cos^{-2}x \end{align*}(4)
\begin{align*} \frac{d}{dx}\sin^{-1}x & =\frac{d\sin x}{dx}\frac{d\sin^{-1}x}{d\sin x}\\ & =\cos x(-\sin^{-2}x)\\ & =-\sin^{-1}x\tan^{-1}x \end{align*}(5)
\begin{align*} \frac{d}{dx}\cos^{-1}x & =\frac{d\cos x}{dx}\frac{d\cos^{-1}x}{d\cos x}\\ & =-\sin x(-\cos^{-2}x)\\ & =\cos^{-1}x\tan x \end{align*}(6)
\begin{align*} \frac{d}{dx}\tan^{-1}x & =\frac{d\tan x}{dx}\frac{d\tan^{-1}x}{d\tan x}\\ & =\cos^{-2}x(-\tan^{-2}x)\\ & =-\sin^{-2}x \end{align*}双曲線関数の微分
(1)
\[ \frac{d}{dx}\sinh x=\cosh x \](2)
\[ \frac{d}{dx}\cosh x=\sinh x \](3)
\[ \frac{d}{dx}\tanh x=\cosh^{-2}x \](4)
\[ \frac{d}{dx}\sinh^{-1}x=-\sinh^{-1}x\tanh^{-1}x \](5)
\[ \frac{d}{dx}\cosh^{-1}x=-\cosh^{-1}x\tanh x \](6)
\[ \frac{d}{dx}\tanh^{-1}x=-\sinh^{-2}x \](1)
\begin{align*} \frac{d}{dx}\sinh x & =-i\frac{d}{dx}\sin(ix)\\ & =\cos(ix)\\ & =\cosh x \end{align*}(2)
\begin{align*} \frac{d}{dx}\cosh x & =\frac{d}{dx}\cos(ix)\\ & =-i\sin(ix)\\ & =\sinh x \end{align*}(3)
\begin{align*} \frac{d}{dx}\tanh x & =-i\frac{d}{dx}\tan(ix)\\ & =\cos^{-2}(ix)\\ & =\cosh^{-2}x \end{align*}(4)
\begin{align*} \frac{d}{dx}\sinh^{-1}x & =i\frac{d}{dx}\sin^{-1}(ix)\\ & =\sin^{-1}(ix)\tan^{-1}(ix)\\ & =-\sinh^{-1}x\tanh^{-1}x \end{align*}(5)
\begin{align*} \frac{d}{dx}\cosh^{-1}x & =\frac{d}{dx}\cos^{-1}(ix)\\ & =i\cos^{-1}(ix)\tan(ix)\\ & =-\cosh^{-1}x\tanh x \end{align*}(6)
\begin{align*} \frac{d}{dx}\tanh^{-1}x & =i\frac{d}{dx}\tan^{-1}(ix)\\ & =\sin^{-2}(ix)\\ & =-\sinh^{-2}x \end{align*}
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三角関数と双曲線関数の対数
\[
\log\sin x=-\log2+\frac{\pi}{2}i-ix-Li_{1}\left(e^{2ix}\right)
\]
巾関数と逆三角関数・逆双曲線関数の積の積分
\[
\int z^{\alpha}\Sin^{\bullet}zdz=\frac{1}{\alpha+1}\left(z^{\alpha+1}\Sin^{\bullet}z-\frac{z^{\alpha+2}}{\alpha+2}F\left(\frac{1}{2},\frac{\alpha}{2}+1;\frac{\alpha}{2}+2;z^{2}\right)\right)+C
\]
三角関数と双曲線関数のn乗積分
\[
\int\sin^{2n+m_{\pm}}xdx=\frac{\Gamma\left(n+\frac{1}{2}+\frac{m_{\pm}}{2}\right)}{\Gamma\left(n+1+\frac{m_{\pm}}{2}\right)}\left\{ -\frac{1}{2}\sum_{k=0}^{n-1}\left(\frac{\Gamma\left(k+1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(k+\frac{3}{2}+\frac{m_{\pm}}{2}\right)}\cos x\sin^{2k+1+m_{\pm}}x\right)+\frac{\Gamma\left(1+\frac{m_{\pm}}{2}\right)}{\Gamma\left(\frac{1}{2}+\frac{m_{\pm}}{2}\right)}\int\sin^{m_{\pm}}xdx\right\}
\]
逆三角関数と逆双曲線関数の積分表示
\[
\sin^{\bullet}x=\int_{0}^{x}\frac{1}{\sqrt{1-t^{2}}}dt
\]