derivative of ((x^2/8+x-1/x)^4)

解

\frac{d}{d x} ((\frac{x ^{2}}{8} + x - \frac{1}{x})^{4})

x (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + 4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + \frac{( x ^{3} + 8 x ^{2} - 8 ) ^{3}}{128 x ^{5}}

解答ステップ

\frac{d}{d x} ((\frac{x ^{2}}{8} + x - \frac{1}{x})^{4})

連鎖法則を適用:

4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} \frac{d}{d x} (\frac{x ^{2}}{8} + x - \frac{1}{x})

= 4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} \frac{d}{d x} (\frac{x ^{2}}{8} + x - \frac{1}{x})

\frac{d}{d x} (\frac{x ^{2}}{8} + x - \frac{1}{x}) = \frac{x}{4} + 1 + \frac{1}{x ^{2}}

= 4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} (\frac{x}{4} + 1 + \frac{1}{x ^{2}})

簡素化

4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} (\frac{x}{4} + 1 + \frac{1}{x ^{2}}) : x (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + 4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + \frac{( x ^{3} + 8 x ^{2} - 8 ) ^{3}}{128 x ^{5}}

= x (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + 4 (\frac{x ^{2}}{8} + x - \frac{1}{x})^{3} + \frac{( x ^{3} + 8 x ^{2} - 8 ) ^{3}}{128 x ^{5}}