(\partial)/(\partial x)(x^{y^x})

解

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

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

解答ステップ

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

y

を定数として扱う

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{\partial}{\partial x} (e^{y^{x} l n (x)})

連鎖法則を適用:

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

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

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

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

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

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