derivative of (2x^2/(3y^2))

解

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

\frac{4 x ( - x \frac{d y}{d x} + y )}{3 y ^{3}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

= \frac{4 x ( - x \frac{d y}{d x} + y )}{3 y ^{3}}