derivative of 4^{9-x^2}

解

\frac{d}{d x} (4^{9 - x^{2}})

- ln (2) \cdot 4^{- x^{2} + 10} x

解答ステップ

\frac{d}{d x} (4^{9 - x^{2}})

指数の規則を適用する:

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

4^{9 - x^{2}} = e^{(9 - x^{2}) l n (4)}

= \frac{d}{d x} (e^{(9 - x^{2}) l n (4)})

連鎖法則を適用:

e^{(9 - x^{2}) l n (4)} \frac{d}{d x} ((9 - x^{2}) ln (4))

= e^{(9 - x^{2}) l n (4)} \frac{d}{d x} ((9 - x^{2}) ln (4))

\frac{d}{d x} ((9 - x^{2}) ln (4)) = - 4 ln (2) x

= e^{(9 - x^{2}) l n (4)} (- 4 ln (2) x)

簡素化

e^{(9 - x^{2}) l n (4)} (- 4 ln (2) x) : - ln (2) \cdot 4^{- x^{2} + 10} x

= - ln (2) \cdot 4^{- x^{2} + 10} x