$3$\displaystyle{{ }^{}{ }^{}{ }^{}{ }^{}{ }^{}{ $ \tan x $ }^{\prime }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}}$

公式の導出はPCサイトを参照のこと。

$3$\displaystyle{\begin{array}{lll}{ $ \tan x $ }^{\prime }& \vspace{6}\\ ={ $ \frac{\hspace{2} \sin x \hspace{2}}{\hspace{2} \cos x \hspace{2}} $ }^{\prime }& \vspace{6}\\ =\frac{\hspace{2} { $ \sin x $ }^{\prime }\cos x- \sin x{ $ \cos x $ }^{\prime } \hspace{2}}{\hspace{2} { $ \cos x $ }^{2} \hspace{2}}& \vspace{6}\\ =\frac{\hspace{2} \cos x\cos x- \sin x $ - \sin x $ \hspace{2}}{\hspace{2} { $ \cos x $ }^{2} \hspace{2}}& \vspace{6}\\ =\frac{\hspace{2} { \cos }^{2}x+{ \sin }^{2}x \hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}& \vspace{6}\\ =\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}& \vspace{6}\\ \end{array}}$

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