derivative f(x)=(x/7+7/x)^7

解

導関数

f (x) = (\frac{x}{7} + \frac{7}{x})^{7}

(\frac{x}{7} + \frac{7}{x})^{6} - \frac{( x ^{2} + 49 ) ^{6}}{2401 x ^{8}}

解答ステップ

\frac{d}{d x} ((\frac{x}{7} + \frac{7}{x})^{7})

連鎖法則を適用:

7 (\frac{x}{7} + \frac{7}{x})^{6} \frac{d}{d x} (\frac{x}{7} + \frac{7}{x})

= 7 (\frac{x}{7} + \frac{7}{x})^{6} \frac{d}{d x} (\frac{x}{7} + \frac{7}{x})

\frac{d}{d x} (\frac{x}{7} + \frac{7}{x}) = \frac{1}{7} - \frac{7}{x ^{2}}

= 7 (\frac{x}{7} + \frac{7}{x})^{6} (\frac{1}{7} - \frac{7}{x ^{2}})

簡素化

7 (\frac{x}{7} + \frac{7}{x})^{6} (\frac{1}{7} - \frac{7}{x ^{2}}) : (\frac{x}{7} + \frac{7}{x})^{6} - \frac{( x ^{2} + 49 ) ^{6}}{2401 x ^{8}}

= (\frac{x}{7} + \frac{7}{x})^{6} - \frac{( x ^{2} + 49 ) ^{6}}{2401 x ^{8}}