derivative y=3arctan(x-sqrt(1+x^2))

解

導関数

y = 3 arctan (x - 1 + x^{2})

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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