derivative of (4(y-x\frac{dy/(dx))}{y^2})

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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