derivative of y^x

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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