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

解

導関数

\frac{3 x}{x ^{2} + 3}

\frac{9}{( x ^{2} + 3 ) x ^{2} + 3}

解答ステップ

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

定数を除去:

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

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

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

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

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

簡素化

3 \cdot \frac{1 \cdot x ^{2} + 3 - \frac{x}{x ^{2} + 3} x}{( x ^{2} + 3 ) ^{2}} : \frac{9}{( x ^{2} + 3 ) x ^{2} + 3}

= \frac{9}{( x ^{2} + 3 ) x ^{2} + 3}