derivative of (cos(x)/(log_{3)(x)})

解

\frac{d}{d x} (\frac{cos ( x )}{lo g _{3} ( x )})

\frac{- x sin ( x ) ln ( x ) - cos ( x )}{ln ( 3 ) x lo g _{3} ( x ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{cos ( x )}{lo g _{3} ( x )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( cos ( x ) ) lo g _{3} ( x ) - \frac{d}{d x} ( lo g _{3} ( x ) ) cos ( x )}{( lo g _{3} ( x ) ) ^{2}}

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

\frac{d}{d x} (lo g_{3} (x)) = \frac{1}{x ln ( 3 )}

= \frac{( - sin ( x ) ) lo g _{3} ( x ) - \frac{1}{x l n ( 3 )} cos ( x )}{( lo g _{3} ( x ) ) ^{2}}

簡素化

\frac{( - sin ( x ) ) lo g _{3} ( x ) - \frac{1}{x l n ( 3 )} cos ( x )}{lo g _{3} ( x ) ^{2}} : \frac{- x sin ( x ) ln ( x ) - cos ( x )}{ln ( 3 ) x lo g _{3} ( x ) ^{2}}

= \frac{- x sin ( x ) ln ( x ) - cos ( x )}{ln ( 3 ) x lo g _{3} ( x ) ^{2}}