derivative f(x)=(120x^2)/(x^2+4)

解

導関数

f (x) = \frac{120 x ^{2}}{x ^{2} + 4}

\frac{960 x}{( x ^{2} + 4 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{120 x ^{2}}{x ^{2} + 4})

定数を除去:

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

= 120 \frac{d}{d x} (\frac{x ^{2}}{x ^{2} + 4})

商の法則を適用:

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

= 120 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( x ^{2} + 4 ) - \frac{d}{d x} ( x ^{2} + 4 ) x ^{2}}{( x ^{2} + 4 ) ^{2}}

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

\frac{d}{d x} (x^{2} + 4) = 2 x

= 120 \cdot \frac{2 x ( x ^{2} + 4 ) - 2 x x ^{2}}{( x ^{2} + 4 ) ^{2}}

簡素化

120 \cdot \frac{2 x ( x ^{2} + 4 ) - 2 x x ^{2}}{( x ^{2} + 4 ) ^{2}} : \frac{960 x}{( x ^{2} + 4 ) ^{2}}

= \frac{960 x}{( x ^{2} + 4 ) ^{2}}