derivative of (x^2/(1+4x))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

\frac{d}{d x} (1 + 4 x) = 4

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

拡張

2 x (1 + 4 x) - 4 x^{2} : 2 x + 4 x^{2}

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