derivative (4x)/(9-cot(x))

解

導関数

\frac{4 x}{9 - cot ( x )}

\frac{4 ( 9 - cot ( x ) - x csc ^{2} ( x ) )}{( 9 - cot ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 x}{9 - cot ( x )})

定数を除去:

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

= 4 \frac{d}{d x} (\frac{x}{9 - cot ( x )})

商の法則を適用:

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

= 4 \cdot \frac{\frac{d x}{d x} ( 9 - cot ( x ) ) - \frac{d}{d x} ( 9 - cot ( x ) ) x}{( 9 - cot ( x ) ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (9 - cot (x)) = csc^{2} (x)

= 4 \cdot \frac{1 \cdot ( 9 - cot ( x ) ) - csc ^{2} ( x ) x}{( 9 - cot ( x ) ) ^{2}}

簡素化

4 \cdot \frac{1 \cdot ( 9 - cot ( x ) ) - csc ^{2} ( x ) x}{( 9 - cot ( x ) ) ^{2}} : \frac{4 ( 9 - cot ( x ) - x csc ^{2} ( x ) )}{( 9 - cot ( x ) ) ^{2}}

= \frac{4 ( 9 - cot ( x ) - x csc ^{2} ( x ) )}{( 9 - cot ( x ) ) ^{2}}