derivative of log_{7}(x^x)

Lösung

\frac{d}{d x} (lo g_{7} (x^{x}))

\frac{ln ( x ) + 1}{ln ( 7 )}

Schritte zur Lösung

\frac{d}{d x} (lo g_{7} (x^{x}))

Wende die log Regel an:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{d}{d x} (\frac{ln ( x ^{x} )}{ln ( 7 )})

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{ln ( 7 )} \frac{d}{d x} (ln (x^{x}))

Wende die Kettenregel an:

\frac{1}{x ^{x}} \frac{d}{d x} (x^{x})

= \frac{1}{x ^{x}} \frac{d}{d x} (x^{x})

\frac{d}{d x} (x^{x}) = x^{x} (ln (x) + 1)

= \frac{1}{ln ( 7 )} \cdot \frac{1}{x ^{x}} x^{x} (ln (x) + 1)

Vereinfache

\frac{1}{ln ( 7 )} \cdot \frac{1}{x ^{x}} x^{x} (ln (x) + 1) : \frac{ln ( x ) + 1}{ln ( 7 )}

= \frac{ln ( x ) + 1}{ln ( 7 )}

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