derivative of (4x/(5-tan(x)))

解

\frac{d}{d x} (\frac{4 x}{5 - tan ( x )})

\frac{4 ( 5 - tan ( x ) + x sec ^{2} ( x ) )}{( 5 - tan ( x ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{4 x}{5 - tan ( x )})

定数を除去:

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

= 4 \frac{d}{d x} (\frac{x}{5 - tan ( 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} ( 5 - tan ( x ) ) - \frac{d}{d x} ( 5 - tan ( x ) ) x}{( 5 - tan ( x ) ) ^{2}}

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

\frac{d}{d x} (5 - tan (x)) = - sec^{2} (x)

= 4 \cdot \frac{1 \cdot ( 5 - tan ( x ) ) - ( - sec ^{2} ( x ) ) x}{( 5 - tan ( x ) ) ^{2}}

簡素化

4 \cdot \frac{1 \cdot ( 5 - tan ( x ) ) - ( - sec ^{2} ( x ) ) x}{( 5 - tan ( x ) ) ^{2}} : \frac{4 ( 5 - tan ( x ) + x sec ^{2} ( x ) )}{( 5 - tan ( x ) ) ^{2}}

= \frac{4 ( 5 - tan ( x ) + x sec ^{2} ( x ) )}{( 5 - tan ( x ) ) ^{2}}