derivative of (x^2+2x^4(x^2+3x^2+7)^3)

Lösung

\frac{d}{d x} ((x^{2} + 2 x)^{4} (x^{2} + 3 x^{2} + 7)^{3})

4 (x^{2} + 2 x)^{3} (2 x + 2) (4 x^{2} + 7)^{3} + 24 x (4 x^{2} + 7)^{2} (x^{2} + 2 x)^{4}

Schritte zur Lösung

\frac{d}{d x} ((x^{2} + 2 x)^{4} (x^{2} + 3 x^{2} + 7)^{3})

Vereinfache

(x^{2} + 2 x)^{4} (x^{2} + 3 x^{2} + 7)^{3} : (x^{2} + 2 x)^{4} (4 x^{2} + 7)^{3}

= \frac{d}{d x} ((x^{2} + 2 x)^{4} (4 x^{2} + 7)^{3})

Wende die Produktregel an:

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

f = (x^{2} + 2 x)^{4}, g = (4 x^{2} + 7)^{3}

= \frac{d}{d x} ((x^{2} + 2 x)^{4}) (4 x^{2} + 7)^{3} + \frac{d}{d x} ((4 x^{2} + 7)^{3}) (x^{2} + 2 x)^{4}

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

\frac{d}{d x} ((4 x^{2} + 7)^{3}) = 24 x (4 x^{2} + 7)^{2}

= 4 (x^{2} + 2 x)^{3} (2 x + 2) (4 x^{2} + 7)^{3} + 24 x (4 x^{2} + 7)^{2} (x^{2} + 2 x)^{4}

t an g e n t y = x^{2} - 5 x + 3, (2, - 3)

\int x \cdot x^{2} - 1 d x

\int \frac{225 - x ^{2}}{x} d x

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

\frac{\partial}{\partial x} (sin (x + e^{y}))