derivative of (sqrt(x)^{3x})

解

\frac{d}{d x} ((x)^{3 x})

3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

解答ステップ

\frac{d}{d x} ((x)^{3 x})

指数の規則を適用する:

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

(x)^{3 x} = e^{3 x l n (x)}

= \frac{d}{d x} (e^{3 x l n (x)})

連鎖法則を適用:

e^{3 x l n (x)} \frac{d}{d x} (3 x ln (x))

= e^{3 x l n (x)} \frac{d}{d x} (3 x ln (x))

\frac{d}{d x} (3 x ln (x)) = 3 (ln (x) + \frac{1}{2})

= e^{3 x l n (x)} \cdot 3 (ln (x) + \frac{1}{2})

簡素化

e^{3 x l n (x)} \cdot 3 (ln (x) + \frac{1}{2}) : 3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= 3 x^{\frac{3 x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})