derivative f(x)=(4+cot(x))/(7-tan(x))

解

導関数

f (x) = \frac{4 + cot ( x )}{7 - tan ( x )}

\frac{- csc ^{2} ( x ) ( 7 - tan ( x ) ) + sec ^{2} ( x ) ( 4 + cot ( x ) )}{( 7 - tan ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 + cot ( x )}{7 - tan ( 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} ( 4 + cot ( x ) ) ( 7 - tan ( x ) ) - \frac{d}{d x} ( 7 - tan ( x ) ) ( 4 + cot ( x ) )}{( 7 - tan ( x ) ) ^{2}}

\frac{d}{d x} (4 + cot (x)) = - csc^{2} (x)

\frac{d}{d x} (7 - tan (x)) = - sec^{2} (x)

= \frac{( - csc ^{2} ( x ) ) ( 7 - tan ( x ) ) - ( - sec ^{2} ( x ) ) ( 4 + cot ( x ) )}{( 7 - tan ( x ) ) ^{2}}

簡素化

= \frac{- csc ^{2} ( x ) ( 7 - tan ( x ) ) + sec ^{2} ( x ) ( 4 + cot ( x ) )}{( 7 - tan ( x ) ) ^{2}}