derivative arctan(9/x)+arctan(3/x)

解

導関数

arctan (\frac{9}{x}) + arctan (\frac{3}{x})

\frac{- 12 x ^{2} - 324}{( x ^{2} + 9 ) ( x ^{2} + 81 )}

解答ステップ

\frac{d}{d x} (arctan (\frac{9}{x}) + arctan (\frac{3}{x}))

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (arctan (\frac{9}{x})) + \frac{d}{d x} (arctan (\frac{3}{x}))

\frac{d}{d x} (arctan (\frac{9}{x})) = - \frac{9}{81 + x ^{2}}

\frac{d}{d x} (arctan (\frac{3}{x})) = - \frac{3}{9 + x ^{2}}

= - \frac{9}{81 + x ^{2}} - \frac{3}{9 + x ^{2}}

簡素化

- \frac{9}{81 + x ^{2}} - \frac{3}{9 + x ^{2}} : \frac{- 12 x ^{2} - 324}{( x ^{2} + 9 ) ( x ^{2} + 81 )}

= \frac{- 12 x ^{2} - 324}{( x ^{2} + 9 ) ( x ^{2} + 81 )}