derivative of x^{3/5}sqrt(x^3+2x-6)

Lösung

\frac{d}{d x} (x^{\frac{3}{5}} x^{3} + 2 x - 6)

\frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

Schritte zur Lösung

\frac{d}{d x} (x^{\frac{3}{5}} x^{3} + 2 x - 6)

Wende die Produktregel an:

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

f = x^{\frac{3}{5}}, g = x^{3} + 2 x - 6

= \frac{d}{d x} (x^{\frac{3}{5}}) x^{3} + 2 x - 6 + \frac{d}{d x} (x^{3} + 2 x - 6) x^{\frac{3}{5}}

\frac{d}{d x} (x^{\frac{3}{5}}) = \frac{3}{5 x ^{\frac{2}{5}}}

\frac{d}{d x} (x^{3} + 2 x - 6) = \frac{3 x ^{2} + 2}{2 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

= \frac{3}{5 x ^{\frac{2}{5}}} x^{3} + 2 x - 6 + \frac{3 x ^{2} + 2}{2 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}} x^{\frac{3}{5}}

Vereinfache

\frac{3}{5 x ^{\frac{2}{5}}} x^{3} + 2 x - 6 + \frac{3 x ^{2} + 2}{2 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}} x^{\frac{3}{5}} : \frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

= \frac{6 x ^{3} + 2 x - 6 ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}} + 5 x ( 3 x ^{2} + 2 )}{10 x ^{\frac{2}{5}} ( x ^{3} + 2 x - 6 ) ^{\frac{1}{2}}}

\frac{d}{d x} (2 x^{2} - 9 x + 10)

in v erse l a pl a ce \frac{2 s - 44}{7 s ^{2} - 14 s + 70}

\frac{d x}{d y} = \frac{( y - x )}{y + x}

a re a y = 2, y = x^{2} + 1

\frac{d}{d x} ((lo g_{2} (x))^{4})