derivative of (10/(sqrt(x^2-4x+1)))

解

\frac{d}{d x} (\frac{10}{x ^{2} - 4 x + 1})

- \frac{5 ( 2 x - 4 )}{( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}}

解答ステップ

\frac{d}{d x} (\frac{10}{x ^{2} - 4 x + 1})

定数を除去:

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

= 10 \frac{d}{d x} (\frac{1}{x ^{2} - 4 x + 1})

指数の規則を適用する:

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

= 10 \frac{d}{d x} ((x^{2} - 4 x + 1)^{- \frac{1}{2}})

連鎖法則を適用:

- \frac{1}{2 ( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} - 4 x + 1)

= - \frac{1}{2 ( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} - 4 x + 1)

\frac{d}{d x} (x^{2} - 4 x + 1) = 2 x - 4

= 10 (- \frac{1}{2 ( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}} (2 x - 4))

簡素化

10 (- \frac{1}{2 ( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}} (2 x - 4)) : - \frac{5 ( 2 x - 4 )}{( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}}

= - \frac{5 ( 2 x - 4 )}{( x ^{2} - 4 x + 1 ) ^{\frac{3}{2}}}