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

解

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

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

解答ステップ

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

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

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

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

拡張

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

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