derivative of 5arctan(x+sqrt(1+x^2))

解

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

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

定数を除去:

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

= 5 \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}}

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

簡素化

5 \cdot \frac{1}{( x + 1 + x ^{2} ) ^{2} + 1} (1 + \frac{x}{1 + x ^{2}}) : \frac{5 ( 1 + x ^{2} + x )}{( 2 x ^{2} + 2 x x ^{2} + 1 + 2 ) 1 + x ^{2}}

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