derivative of x^{5^x}

Lösung

\frac{d}{d x} (x^{5^{x}})

x^{5^{x}} (ln (5) \cdot 5^{x} ln (x) + \frac{5 ^{x}}{x})

Schritte zur Lösung

\frac{d}{d x} (x^{5^{x}})

Wende Exponentenregel an:

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

x^{5^{x}} = e^{5^{x} l n (x)}

= \frac{d}{d x} (e^{5^{x} l n (x)})

Wende die Kettenregel an:

e^{5^{x} l n (x)} \frac{d}{d x} (5^{x} ln (x))

= e^{5^{x} l n (x)} \frac{d}{d x} (5^{x} ln (x))

\frac{d}{d x} (5^{x} ln (x)) = ln (5) \cdot 5^{x} ln (x) + \frac{5 ^{x}}{x}

= e^{5^{x} l n (x)} (ln (5) \cdot 5^{x} ln (x) + \frac{5 ^{x}}{x})

e^{5^{x} l n (x)} = x^{5^{x}}

= x^{5^{x}} (ln (5) \cdot 5^{x} ln (x) + \frac{5 ^{x}}{x})

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