derivative of log_{10}(cos(x))

解

\frac{d}{d x} (lo g_{10} (cos (x)))

- \frac{tan ( x )}{ln ( 10 )}

解答ステップ

\frac{d}{d x} (lo g_{10} (cos (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 ( cos ( x ) )}{ln ( 10 )})

定数を除去:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (cos (x)))

連鎖法則を適用:

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

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

\frac{d}{d x} (cos (x)) = - sin (x)

= \frac{1}{ln ( 10 )} \cdot \frac{1}{cos ( x )} (- sin (x))

簡素化

\frac{1}{ln ( 10 )} \cdot \frac{1}{cos ( x )} (- sin (x)) : - \frac{tan ( x )}{ln ( 10 )}

= - \frac{tan ( x )}{ln ( 10 )}