derivative of 1/((1+x^2sqrt(1+x^2)))

解

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

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

解答ステップ

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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