derivative of x^y-y^x

解答

\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}

求解步骤

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

使用微分加减法定则:

(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}

y^{^{'}} - y = x^{2}

x \to 7 lim (3 x - 5)

x \to 0 lim (\frac{2 x ^{2} + 5 x + 7}{x})

\frac{d}{d x} (cos (sin (5 x)))

x y^{^{'}} + y = 40 x