derivative of (sqrt(x)^{7x})

解

\frac{d}{d x} ((x)^{7 x})

7 x^{\frac{7 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

解答ステップ

\frac{d}{d x} ((x)^{7 x})

指数の規則を適用する:

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

(x)^{7 x} = e^{7 x l n (x)}

= \frac{d}{d x} (e^{7 x l n (x)})

連鎖法則を適用:

e^{7 x l n (x)} \frac{d}{d x} (7 x ln (x))

= e^{7 x l n (x)} \frac{d}{d x} (7 x ln (x))

\frac{d}{d x} (7 x ln (x)) = 7 (ln (x) + \frac{1}{2})

= e^{7 x l n (x)} \cdot 7 (ln (x) + \frac{1}{2})

簡素化

e^{7 x l n (x)} \cdot 7 (ln (x) + \frac{1}{2}) : 7 x^{\frac{7 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= 7 x^{\frac{7 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})