derivative of x^{y^x}

Lösung

\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})

Schritte zur Lösung

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

Wende Exponentenregel an:

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)})

Wende die Kettenregel an:

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})

\int \frac{3}{x ^{2} + 4} d x

\int x^{4} x^{5} + 3 d x

\frac{\partial}{\partial y} (2 e^{\frac{x}{2} + y} - y)

\frac{d}{d x} (\frac{x ^{2} ( x + 2 ) ^{4}}{( 2 x ^{2} - 1 ) ^{3}})

l a pl a ce t r an s f or m sin (t - 8)