derivative of log_{5}(tan(x))

解

\frac{d}{d x} (lo g_{5} (tan (x)))

\frac{sec ^{2} ( x )}{ln ( 5 ) tan ( x )}

解答ステップ

\frac{d}{d x} (lo g_{5} (tan (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 ( tan ( x ) )}{ln ( 5 )})

定数を除去:

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

= \frac{1}{ln ( 5 )} \frac{d}{d x} (ln (tan (x)))

連鎖法則を適用:

\frac{1}{tan ( x )} \frac{d}{d x} (tan (x))

= \frac{1}{tan ( x )} \frac{d}{d x} (tan (x))

\frac{d}{d x} (tan (x)) = sec^{2} (x)

= \frac{1}{ln ( 5 )} \cdot \frac{1}{tan ( x )} sec^{2} (x)

簡素化

\frac{1}{ln ( 5 )} \cdot \frac{1}{tan ( x )} sec^{2} (x) : \frac{sec ^{2} ( x )}{ln ( 5 ) tan ( x )}

= \frac{sec ^{2} ( x )}{ln ( 5 ) tan ( x )}