derivative of (2x^3/(sqrt(x^2-3)))

解

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

\frac{2 ( 2 x ^{4} - 9 x ^{2} )}{( x ^{2} - 3 ) x ^{2} - 3}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

2 \cdot \frac{3 x ^{2} x ^{2} - 3 - \frac{x}{x ^{2} - 3} x ^{3}}{( x ^{2} - 3 ) ^{2}} : \frac{2 ( 2 x ^{4} - 9 x ^{2} )}{( x ^{2} - 3 ) x ^{2} - 3}

= \frac{2 ( 2 x ^{4} - 9 x ^{2} )}{( x ^{2} - 3 ) x ^{2} - 3}