derivative of (y^2-3x^2/(y^4))

解

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

\frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}

解答ステップ

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

商の法則を適用:

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

= \frac{\frac{d}{d x} ( y ^{2} - 3 x ^{2} ) y ^{4} - \frac{d}{d x} ( y ^{4} ) ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}}

\frac{d}{d x} (y^{2} - 3 x^{2}) = 2 y \frac{d y}{d x} - 6 x

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

= \frac{( 2 y \frac{d y}{d x} - 6 x ) y ^{4} - 4 y ^{3} \frac{d y}{d x} ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}}

簡素化

\frac{( 2 y \frac{d y}{d x} - 6 x ) y ^{4} - 4 y ^{3} \frac{d y}{d x} ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}} : \frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}

= \frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}