derivative of log_{6}(cot(8x))

解

\frac{d}{d x} (lo g_{6} (cot (8 x)))

- \frac{8 csc ^{2} ( 8 x )}{ln ( 6 ) cot ( 8 x )}

解答ステップ

\frac{d}{d x} (lo g_{6} (cot (8 x)))

対数の規則を適用する:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( cot ( 8 x ) )}{ln ( 6 )})

定数を除去:

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

= \frac{1}{ln ( 6 )} \frac{d}{d x} (ln (cot (8 x)))

連鎖法則を適用:

\frac{1}{cot ( 8 x )} \frac{d}{d x} (cot (8 x))

= \frac{1}{cot ( 8 x )} \frac{d}{d x} (cot (8 x))

\frac{d}{d x} (cot (8 x)) = - csc^{2} (8 x) \cdot 8

= \frac{1}{ln ( 6 )} \cdot \frac{1}{cot ( 8 x )} (- csc^{2} (8 x) \cdot 8)

簡素化

\frac{1}{ln ( 6 )} \cdot \frac{1}{cot ( 8 x )} (- csc^{2} (8 x) \cdot 8) : - \frac{8 csc ^{2} ( 8 x )}{ln ( 6 ) cot ( 8 x )}

= - \frac{8 csc ^{2} ( 8 x )}{ln ( 6 ) cot ( 8 x )}