derivative of (4+csc(x)/(8-csc(x)))

解

\frac{d}{d x} (\frac{4 + csc ( x )}{8 - csc ( x )})

- \frac{12 cot ( x ) csc ( x )}{( 8 - csc ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 + csc ( x )}{8 - csc ( 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 + csc ( x ) ) ( 8 - csc ( x ) ) - \frac{d}{d x} ( 8 - csc ( x ) ) ( 4 + csc ( x ) )}{( 8 - csc ( x ) ) ^{2}}

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

\frac{d}{d x} (8 - csc (x)) = cot (x) csc (x)

= \frac{( - cot ( x ) csc ( x ) ) ( 8 - csc ( x ) ) - cot ( x ) csc ( x ) ( 4 + csc ( x ) )}{( 8 - csc ( x ) ) ^{2}}

簡素化

\frac{( - cot ( x ) csc ( x ) ) ( 8 - csc ( x ) ) - cot ( x ) csc ( x ) ( 4 + csc ( x ) )}{( 8 - csc ( x ) ) ^{2}} : - \frac{12 cot ( x ) csc ( x )}{( 8 - csc ( x ) ) ^{2}}

= - \frac{12 cot ( x ) csc ( x )}{( 8 - csc ( x ) ) ^{2}}