derivative y=((2x^2)/(6-x))^3

解

導関数

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

\frac{24 x ^{4} ( 12 x - x ^{2} )}{( 6 - x ) ^{4}}

解答ステップ

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

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{2 x ^{2}}{6 - x}) = \frac{2 ( 12 x - x ^{2} )}{( 6 - x ) ^{2}}

= 3 (\frac{2 x ^{2}}{6 - x})^{2} \frac{2 ( 12 x - x ^{2} )}{( 6 - x ) ^{2}}

簡素化

3 (\frac{2 x ^{2}}{6 - x})^{2} \frac{2 ( 12 x - x ^{2} )}{( 6 - x ) ^{2}} : \frac{24 x ^{4} ( 12 x - x ^{2} )}{( 6 - x ) ^{4}}

= \frac{24 x ^{4} ( 12 x - x ^{2} )}{( 6 - x ) ^{4}}