derivative of x^{1/2}ln(x)

Lösung

\frac{d}{d x} (x^{\frac{1}{2}} ln (x))

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

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{1}{2}} ln (x))

Wende die Produktregel an:

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

f = x^{\frac{1}{2}}, g = ln (x)

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

\frac{d}{d x} (x^{\frac{1}{2}}) = \frac{1}{2 x ^{\frac{1}{2}}}

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

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

Vereinfache

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

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