derivative of x^{y^x}

解

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

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

解答ステップ

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

指数の規則を適用する:

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

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

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

連鎖法則を適用:

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

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

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

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

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

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