derivative of ((x^2/4+5x-1/x)^3)

解

\frac{d}{d x} ((\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{3})

\frac{3 ( x ^{9} + 50 x ^{8} + 800 x ^{7} + 3994 x ^{6} - 160 x ^{5} - 160 x ^{2} + 32 )}{32 x ^{4}} - 75

解答ステップ

\frac{d}{d x} ((\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{3})

連鎖法則を適用:

3 (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{2} \frac{d}{d x} (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})

= 3 (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{2} \frac{d}{d x} (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})

\frac{d}{d x} (\frac{x ^{2}}{4} + 5 x - \frac{1}{x}) = \frac{x}{2} + 5 + \frac{1}{x ^{2}}

= 3 (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{2} (\frac{x}{2} + 5 + \frac{1}{x ^{2}})

簡素化

3 (\frac{x ^{2}}{4} + 5 x - \frac{1}{x})^{2} (\frac{x}{2} + 5 + \frac{1}{x ^{2}}) : \frac{3 ( x ^{9} + 50 x ^{8} + 800 x ^{7} + 3994 x ^{6} - 160 x ^{5} - 160 x ^{2} + 32 )}{32 x ^{4}} - 75

= \frac{3 ( x ^{9} + 50 x ^{8} + 800 x ^{7} + 3994 x ^{6} - 160 x ^{5} - 160 x ^{2} + 32 )}{32 x ^{4}} - 75