derivative of (3x/(sqrt(2x+3)))

解

\frac{d}{d x} (\frac{3 x}{2 x + 3})

\frac{3 ( x + 3 )}{( 2 x + 3 ) 2 x + 3}

解答ステップ

\frac{d}{d x} (\frac{3 x}{2 x + 3})

定数を除去:

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

= 3 \frac{d}{d x} (\frac{x}{2 x + 3})

商の法則を適用:

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

= 3 \cdot \frac{\frac{d x}{d x} 2 x + 3 - \frac{d}{d x} ( 2 x + 3 ) x}{( 2 x + 3 ) ^{2}}

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

\frac{d}{d x} (2 x + 3) = \frac{1}{2 x + 3}

= 3 \cdot \frac{1 \cdot 2 x + 3 - \frac{1}{2 x + 3} x}{( 2 x + 3 ) ^{2}}

簡素化

3 \cdot \frac{1 \cdot 2 x + 3 - \frac{1}{2 x + 3} x}{( 2 x + 3 ) ^{2}} : \frac{3 ( x + 3 )}{( 2 x + 3 ) 2 x + 3}

= \frac{3 ( x + 3 )}{( 2 x + 3 ) 2 x + 3}