derivative of (2^x*x^{-2}^{sqrt(3x-1)})

解

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

\frac{2 ^{x 3 x - 1 - 1} ( 9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 6 x ln ( x ) - 12 x + 4 )}{x ^{2 3 x - 1 + 1} 3 x - 1}

解答ステップ

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

指数の規則を適用する:

a^{b} = e^{b l n (a)}

(2^{x} x^{- 2})^{3 x - 1} = e^{3 x - 1 l n (2^{x} x^{- 2})}

= \frac{d}{d x} (e^{3 x - 1 l n (2^{x} x^{- 2})})

連鎖法則を適用:

e^{3 x - 1 l n (2^{x} x^{- 2})} \frac{d}{d x} (3 x - 1 ln (2^{x} x^{- 2}))

= e^{3 x - 1 l n (2^{x} x^{- 2})} \frac{d}{d x} (3 x - 1 ln (2^{x} x^{- 2}))

\frac{d}{d x} (3 x - 1 ln (2^{x} x^{- 2})) = \frac{9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 3 x ln ( x ^{2} ) - 12 x + 4}{2 x 3 x - 1}

= e^{3 x - 1 l n (2^{x} x^{- 2})} \frac{9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 3 x ln ( x ^{2} ) - 12 x + 4}{2 x 3 x - 1}

簡素化

e^{3 x - 1 l n (2^{x} x^{- 2})} \frac{9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 3 x ln ( x ^{2} ) - 12 x + 4}{2 x 3 x - 1} : \frac{2 ^{x 3 x - 1 - 1} ( 9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 6 x ln ( x ) - 12 x + 4 )}{x ^{2 3 x - 1 + 1} 3 x - 1}

= \frac{2 ^{x 3 x - 1 - 1} ( 9 ln ( 2 ) x ^{2} - 2 ln ( 2 ) x - 6 x ln ( x ) - 12 x + 4 )}{x ^{2 3 x - 1 + 1} 3 x - 1}