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微分

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