derivative of-5^{8x^2-1}

解

\frac{d}{d x} (- 5^{8 x^{2} - 1})

- 16 ln (5) \cdot 5^{8 x^{2} - 1} x

解答ステップ

\frac{d}{d x} (- 5^{8 x^{2} - 1})

定数を除去:

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

= - \frac{d}{d x} (5^{8 x^{2} - 1})

指数の規則を適用する:

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

5^{8 x^{2} - 1} = e^{(8 x^{2} - 1) l n (5)}

= - \frac{d}{d x} (e^{(8 x^{2} - 1) l n (5)})

連鎖法則を適用:

e^{(8 x^{2} - 1) l n (5)} \frac{d}{d x} ((8 x^{2} - 1) ln (5))

= e^{(8 x^{2} - 1) l n (5)} \frac{d}{d x} ((8 x^{2} - 1) ln (5))

\frac{d}{d x} ((8 x^{2} - 1) ln (5)) = 16 ln (5) x

= - e^{(8 x^{2} - 1) l n (5)} \cdot 16 ln (5) x

e^{(8 x^{2} - 1) l n (5)} = 5^{8 x^{2} - 1}

= - 16 ln (5) \cdot 5^{8 x^{2} - 1} x