derivative of log_{4}(sec(x))

解

\frac{d}{d x} (lo g_{4} (sec (x)))

\frac{tan ( x )}{2 ln ( 2 )}

解答ステップ

\frac{d}{d x} (lo g_{4} (sec (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 ( sec ( x ) )}{ln ( 4 )})

定数を除去:

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

= \frac{1}{ln ( 4 )} \frac{d}{d x} (ln (sec (x)))

連鎖法則を適用:

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

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

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

= \frac{1}{ln ( 4 )} \cdot \frac{1}{sec ( x )} sec (x) tan (x)

簡素化

\frac{1}{ln ( 4 )} \cdot \frac{1}{sec ( x )} sec (x) tan (x) : \frac{tan ( x )}{2 ln ( 2 )}

= \frac{tan ( x )}{2 ln ( 2 )}