derivative of-1/3 ln((tan(x/2-3)/3))

解

\frac{d}{d x} (- \frac{1}{3} ln (\frac{tan ( \frac{x}{2} ) - 3}{3}))

- \frac{sec ^{2} ( \frac{x}{2} )}{6 ( tan ( \frac{x}{2} ) - 3 )}

解答ステップ

\frac{d}{d x} (- \frac{1}{3} ln (\frac{tan ( \frac{x}{2} ) - 3}{3}))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= - \frac{1}{3} \frac{d}{d x} (ln (\frac{tan ( \frac{x}{2} ) - 3}{3}))

連鎖法則を適用:

\frac{1}{\frac{t a n ( \frac{x}{2} ) - 3}{3}} \frac{d}{d x} (\frac{tan ( \frac{x}{2} ) - 3}{3})

= \frac{1}{\frac{t a n ( \frac{x}{2} ) - 3}{3}} \frac{d}{d x} (\frac{tan ( \frac{x}{2} ) - 3}{3})

\frac{d}{d x} (\frac{tan ( \frac{x}{2} ) - 3}{3}) = \frac{1}{6} sec^{2} (\frac{x}{2})

= - \frac{1}{3} \cdot \frac{1}{\frac{t a n ( \frac{x}{2} ) - 3}{3}} \cdot \frac{1}{6} sec^{2} (\frac{x}{2})

簡素化

- \frac{1}{3} \cdot \frac{1}{\frac{t a n ( \frac{x}{2} ) - 3}{3}} \cdot \frac{1}{6} sec^{2} (\frac{x}{2}) : - \frac{sec ^{2} ( \frac{x}{2} )}{6 ( tan ( \frac{x}{2} ) - 3 )}

= - \frac{sec ^{2} ( \frac{x}{2} )}{6 ( tan ( \frac{x}{2} ) - 3 )}