derivative of (3x^2/(x^3+1))

解

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

\frac{3 ( - x ^{4} + 2 x )}{( x ^{3} + 1 ) ^{2}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

= \frac{3 ( - x ^{4} + 2 x )}{( x ^{3} + 1 ) ^{2}}