derivative of (5x/(sqrt(x^2+6)))

解

\frac{d}{d x} (\frac{5 x}{x ^{2} + 6})

\frac{30}{( x ^{2} + 6 ) x ^{2} + 6}

解答ステップ

\frac{d}{d x} (\frac{5 x}{x ^{2} + 6})

定数を除去:

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

= 5 \frac{d}{d x} (\frac{x}{x ^{2} + 6})

商の法則を適用:

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

= 5 \cdot \frac{\frac{d x}{d x} x ^{2} + 6 - \frac{d}{d x} ( x ^{2} + 6 ) x}{( x ^{2} + 6 ) ^{2}}

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

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

= 5 \cdot \frac{1 \cdot x ^{2} + 6 - \frac{x}{x ^{2} + 6} x}{( x ^{2} + 6 ) ^{2}}

簡素化

5 \cdot \frac{1 \cdot x ^{2} + 6 - \frac{x}{x ^{2} + 6} x}{( x ^{2} + 6 ) ^{2}} : \frac{30}{( x ^{2} + 6 ) x ^{2} + 6}

= \frac{30}{( x ^{2} + 6 ) x ^{2} + 6}