derivative of \sqrt[4]{(x^2+1/(x^2-1)})

解

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

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

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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