derivative x^x(log_{10}(x)+1)

Lösung

ableitung von

x^{x} (lo g_{10} (x) + 1)

\frac{x ^{x - 1} + x ^{x} ln ^{2} ( x ) + ln ( 10 ) x ^{x} ln ( x ) + x ^{x} ln ( x ) + ln ( 10 ) x ^{x}}{ln ( 10 )}

Schritte zur Lösung

\frac{d}{d x} (x^{x} (lo g_{10} (x) + 1))

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{x}, g = lo g_{10} (x) + 1

= \frac{d}{d x} (x^{x}) (lo g_{10} (x) + 1) + \frac{d}{d x} (lo g_{10} (x) + 1) x^{x}

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

\frac{d}{d x} (lo g_{10} (x) + 1) = \frac{1}{ln ( 10 ) x}

= x^{x} (ln (x) + 1) (lo g_{10} (x) + 1) + \frac{1}{ln ( 10 ) x} x^{x}

Vereinfache

x^{x} (ln (x) + 1) (lo g_{10} (x) + 1) + \frac{1}{ln ( 10 ) x} x^{x} : \frac{x ^{x - 1} + x ^{x} ln ^{2} ( x ) + ln ( 10 ) x ^{x} ln ( x ) + x ^{x} ln ( x ) + ln ( 10 ) x ^{x}}{ln ( 10 )}

= \frac{x ^{x - 1} + x ^{x} ln ^{2} ( x ) + ln ( 10 ) x ^{x} ln ( x ) + x ^{x} ln ( x ) + ln ( 10 ) x ^{x}}{ln ( 10 )}

d er i v a t i v e f (x) = x^{5} + x^{2} - 2 x + 1

\frac{d}{d t} (2 t (\frac{n}{4}) + c)

d er i v a t i v e ln (tan (\frac{x}{2}))

d er i v a t i v e 4 e^{2} x

d er i v a t i v e \frac{( x ^{2} )}{10}