derivative of (3x^2+ln(x)*(5x^3)^x)

解

\frac{d}{d x} ((3 x^{2} + ln (x)) \cdot (5 x^{3})^{x})

(6 x + \frac{1}{x}) (5 x^{3})^{x} + (ln (5 x^{3}) + 3) (5 x^{3})^{x} (3 x^{2} + ln (x))

解答ステップ

\frac{d}{d x} ((3 x^{2} + ln (x)) (5 x^{3})^{x})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = 3 x^{2} + ln (x), g = (5 x^{3})^{x}

= \frac{d}{d x} (3 x^{2} + ln (x)) (5 x^{3})^{x} + \frac{d}{d x} ((5 x^{3})^{x}) (3 x^{2} + ln (x))

\frac{d}{d x} (3 x^{2} + ln (x)) = 6 x + \frac{1}{x}

\frac{d}{d x} ((5 x^{3})^{x}) = (ln (5 x^{3}) + 3) (5 x^{3})^{x}

= (6 x + \frac{1}{x}) (5 x^{3})^{x} + (ln (5 x^{3}) + 3) (5 x^{3})^{x} (3 x^{2} + ln (x))