derivative e^{3x}*sin(ln(x))

解

導関数

e^{3 x} \cdot sin (ln (x))

3 e^{3 x} sin (ln (x)) + \frac{e ^{3 x} cos ( ln ( x ) )}{x}

解答ステップ

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

積の法則を適用:

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

f = e^{3 x}, g = sin (ln (x))

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

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

\frac{d}{d x} (sin (ln (x))) = \frac{cos ( ln ( x ) )}{x}

= e^{3 x} \cdot 3 sin (ln (x)) + \frac{cos ( ln ( x ) )}{x} e^{3 x}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 3 e^{3 x} sin (ln (x)) + \frac{e ^{3 x} cos ( ln ( x ) )}{x}