derivative of (c^3-x^3/(c^3+x^3))

解

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

- \frac{6 c ^{3} x ^{2}}{( c ^{3} + x ^{3} ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

\frac{( - 3 x ^{2} ) ( c ^{3} + x ^{3} ) - 3 x ^{2} ( c ^{3} - x ^{3} )}{( c ^{3} + x ^{3} ) ^{2}} : - \frac{6 c ^{3} x ^{2}}{( c ^{3} + x ^{3} ) ^{2}}

= - \frac{6 c ^{3} x ^{2}}{( c ^{3} + x ^{3} ) ^{2}}