derivative of (x^2/((x+2i)^2))

解

\frac{d}{d x} (\frac{x ^{2}}{( x + 2 i ) ^{2}})

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

解答ステップ

\frac{d}{d x} (\frac{x ^{2}}{( x + 2 i ) ^{2}})

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} ) ( x + 2 i ) ^{2} - \frac{d}{d x} ( ( x + 2 i ) ^{2} ) x ^{2}}{( ( x + 2 i ) ^{2} ) ^{2}}

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

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

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

簡素化

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

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