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

解

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

\frac{5 ( x ^{4} + 1 ) ^{4} ( 2 x ^{5} + 4 x ^{3} - 2 x )}{( x ^{2} + 1 ) ^{6}}

解答ステップ

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

連鎖法則を適用:

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

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

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

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

簡素化

5 (\frac{x ^{4} + 1}{x ^{2} + 1})^{4} \frac{2 x ^{5} + 4 x ^{3} - 2 x}{( x ^{2} + 1 ) ^{2}} : \frac{5 ( x ^{4} + 1 ) ^{4} ( 2 x ^{5} + 4 x ^{3} - 2 x )}{( x ^{2} + 1 ) ^{6}}

= \frac{5 ( x ^{4} + 1 ) ^{4} ( 2 x ^{5} + 4 x ^{3} - 2 x )}{( x ^{2} + 1 ) ^{6}}