derivative (sqrt(x))^{5x}

解

導関数

(x)^{5 x}

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

解答ステップ

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

指数の規則を適用する:

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

(x)^{5 x} = e^{5 x l n (x)}

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

連鎖法則を適用:

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

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

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

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

e^{5 x l n (x)} = (x)^{5 x}

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