derivative y=(2x+1)^{2/3}(3x-4)^{1/2}

Lösung

ableitung von

y = (2 x + 1)^{\frac{2}{3}} (3 x - 4)^{\frac{1}{2}}

\frac{42 x - 23}{6 ( 2 x + 1 ) ^{\frac{1}{3}} ( 3 x - 4 ) ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} ((2 x + 1)^{\frac{2}{3}} (3 x - 4)^{\frac{1}{2}})

Wende die Produktregel an:

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

f = (2 x + 1)^{\frac{2}{3}}, g = (3 x - 4)^{\frac{1}{2}}

= \frac{d}{d x} ((2 x + 1)^{\frac{2}{3}}) (3 x - 4)^{\frac{1}{2}} + \frac{d}{d x} ((3 x - 4)^{\frac{1}{2}}) (2 x + 1)^{\frac{2}{3}}

\frac{d}{d x} ((2 x + 1)^{\frac{2}{3}}) = \frac{4}{3 ( 2 x + 1 ) ^{\frac{1}{3}}}

\frac{d}{d x} ((3 x - 4)^{\frac{1}{2}}) = \frac{3}{2 ( 3 x - 4 ) ^{\frac{1}{2}}}

= \frac{4}{3 ( 2 x + 1 ) ^{\frac{1}{3}}} (3 x - 4)^{\frac{1}{2}} + \frac{3}{2 ( 3 x - 4 ) ^{\frac{1}{2}}} (2 x + 1)^{\frac{2}{3}}

Vereinfache

\frac{4}{3 ( 2 x + 1 ) ^{\frac{1}{3}}} (3 x - 4)^{\frac{1}{2}} + \frac{3}{2 ( 3 x - 4 ) ^{\frac{1}{2}}} (2 x + 1)^{\frac{2}{3}} : \frac{42 x - 23}{6 ( 2 x + 1 ) ^{\frac{1}{3}} ( 3 x - 4 ) ^{\frac{1}{2}}}

= \frac{42 x - 23}{6 ( 2 x + 1 ) ^{\frac{1}{3}} ( 3 x - 4 ) ^{\frac{1}{2}}}

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