derivative of y^{3x}

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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