derivative of 5^{-1/x}

解

\frac{d}{d x} (5^{- \frac{1}{x}})

\frac{ln ( 5 ) \cdot 5 ^{- \frac{1}{x}}}{x ^{2}}

解答ステップ

\frac{d}{d x} (5^{- \frac{1}{x}})

指数の規則を適用する:

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

5^{- \frac{1}{x}} = e^{(- \frac{1}{x}) l n (5)}

= \frac{d}{d x} (e^{(- \frac{1}{x}) l n (5)})

連鎖法則を適用:

e^{(- \frac{1}{x}) l n (5)} \frac{d}{d x} ((- \frac{1}{x}) ln (5))

= e^{(- \frac{1}{x}) l n (5)} \frac{d}{d x} ((- \frac{1}{x}) ln (5))

\frac{d}{d x} ((- \frac{1}{x}) ln (5)) = \frac{ln ( 5 )}{x ^{2}}

= e^{(- \frac{1}{x}) l n (5)} \frac{ln ( 5 )}{x ^{2}}

簡素化

e^{(- \frac{1}{x}) l n (5)} \frac{ln ( 5 )}{x ^{2}} : \frac{ln ( 5 ) \cdot 5 ^{- \frac{1}{x}}}{x ^{2}}

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