derivative of x^2sqrt(x^3+a^3)

Lösung

\frac{d}{d x} (x^{2} x^{3} + a^{3})

\frac{7 x ^{4} + 4 a ^{3} x}{2 x ^{3} + a ^{3}}

Schritte zur Lösung

\frac{d}{d x} (x^{2} x^{3} + a^{3})

Wende die Produktregel an:

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

f = x^{2}, g = x^{3} + a^{3}

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

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

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

= 2 x x^{3} + a^{3} + \frac{3 x ^{2}}{2 x ^{3} + a ^{3}} x^{2}

Vereinfache

2 x x^{3} + a^{3} + \frac{3 x ^{2}}{2 x ^{3} + a ^{3}} x^{2} : \frac{7 x ^{4} + 4 a ^{3} x}{2 x ^{3} + a ^{3}}

= \frac{7 x ^{4} + 4 a ^{3} x}{2 x ^{3} + a ^{3}}

\int \frac{3 x ^{3} + 3 x + 2}{x ^{3} + x} d x

t an g e n t f (x) = \frac{1}{x ^{4}}, at x = 3

\frac{\partial}{\partial z} (x e^{z})

d er i v a t i v e \frac{1}{x + 9}

\int \frac{e ^{\frac{x}{2}}}{x} d x