derivative x^{1/2}log_{10}(x)

Lösung

ableitung von

x^{\frac{1}{2}} lo g_{10} (x)

\frac{lo g _{10} ( x )}{2 x ^{\frac{1}{2}}} + \frac{1}{ln ( 10 ) x ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{1}{2}} lo g_{10} (x))

Wende die Produktregel an:

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

f = x^{\frac{1}{2}}, g = lo g_{10} (x)

= \frac{d}{d x} (x^{\frac{1}{2}}) lo g_{10} (x) + \frac{d}{d x} (lo g_{10} (x)) x^{\frac{1}{2}}

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

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

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

Vereinfache

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

= \frac{lo g _{10} ( x )}{2 x ^{\frac{1}{2}}} + \frac{1}{ln ( 10 ) x ^{\frac{1}{2}}}

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