derivative of (3x+6/(2sqrt(x+3)))

解

\frac{d}{d x} (\frac{3 x + 6}{2 x + 3})

\frac{3 x + 12}{4 ( x + 3 ) x + 3}

解答ステップ

\frac{d}{d x} (\frac{3 x + 6}{2 x + 3})

定数を除去:

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

= \frac{1}{2} \frac{d}{d x} (\frac{3 x + 6}{x + 3})

商の法則を適用:

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

= \frac{1}{2} \cdot \frac{\frac{d}{d x} ( 3 x + 6 ) x + 3 - \frac{d}{d x} ( x + 3 ) ( 3 x + 6 )}{( x + 3 ) ^{2}}

\frac{d}{d x} (3 x + 6) = 3

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

= \frac{1}{2} \cdot \frac{3 x + 3 - \frac{1}{2 x + 3} ( 3 x + 6 )}{( x + 3 ) ^{2}}

簡素化

\frac{1}{2} \cdot \frac{3 x + 3 - \frac{1}{2 x + 3} ( 3 x + 6 )}{( x + 3 ) ^{2}} : \frac{3 x + 12}{4 ( x + 3 ) x + 3}

= \frac{3 x + 12}{4 ( x + 3 ) x + 3}