derivative of x^y+y^x

解

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

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

解答ステップ

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

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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