derivative of e^{4x}ln(y)

解

\frac{d}{d x} (e^{4 x} ln (y))

4 e^{4 x} ln (y) + \frac{e ^{4 x} \frac{d y}{d x}}{y}

解答ステップ

\frac{d}{d x} (e^{4 x} ln (y))

積の法則を適用:

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

f = e^{4 x}, g = ln (y)

= \frac{d}{d x} (e^{4 x}) ln (y) + \frac{d}{d x} (ln (y)) e^{4 x}

\frac{d}{d x} (e^{4 x}) = e^{4 x} \cdot 4

\frac{d}{d x} (ln (y)) = \frac{\frac{d y}{d x}}{y}

= e^{4 x} \cdot 4 ln (y) + \frac{\frac{d y}{d x}}{y} e^{4 x}

分数を乗じる:

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

= 4 e^{4 x} ln (y) + \frac{e ^{4 x} \frac{d y}{d x}}{y}