derivative of arctan((3x-8/(x^3)))

解

\frac{d}{d x} (arctan (\frac{3 x - 8}{x ^{3}}))

\frac{6 x ^{2} ( - x + 4 )}{( 3 x - 8 ) ^{2} + x ^{6}}

解答ステップ

\frac{d}{d x} (arctan (\frac{3 x - 8}{x ^{3}}))

連鎖法則を適用:

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

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

\frac{d}{d x} (\frac{3 x - 8}{x ^{3}}) = \frac{6 ( - x + 4 )}{x ^{4}}

= \frac{1}{( \frac{3 x - 8}{x ^{3}} ) ^{2} + 1} \cdot \frac{6 ( - x + 4 )}{x ^{4}}

簡素化

\frac{1}{( \frac{3 x - 8}{x ^{3}} ) ^{2} + 1} \cdot \frac{6 ( - x + 4 )}{x ^{4}} : \frac{6 x ^{2} ( - x + 4 )}{( 3 x - 8 ) ^{2} + x ^{6}}

= \frac{6 x ^{2} ( - x + 4 )}{( 3 x - 8 ) ^{2} + x ^{6}}