derivative of (3x^2^{2x}*(x^2+2x)^2)

Lösung

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

2 ln (3) \cdot 9^{x} x^{4 x} (x^{2} + 2 x)^{2} + 9^{x} (4 x^{4 x + 4} ln (x) + 16 x^{4 x + 3} ln (x) + 16 x^{4 x + 2} ln (x) + 4 x^{4 x + 4} + 20 x^{4 x + 3} + 28 x^{4 x + 2} + 8 x^{4 x + 1})

Schritte zur Lösung

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

Vereinfache

(3 x^{2})^{2 x} (x^{2} + 2 x)^{2} : 9^{x} (x^{2})^{2 x} (x^{2} + 2 x)^{2}

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

Wende die Produktregel an:

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

f = 9^{x}, g = (x^{2})^{2 x} (x^{2} + 2 x)^{2}

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

\frac{d}{d x} (9^{x}) = 2 ln (3) \cdot 9^{x}

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

= 2 ln (3) \cdot 9^{x} (x^{2})^{2 x} (x^{2} + 2 x)^{2} + (2 x^{4 x} (2 ln (x) + 2) (x^{2} + 2 x)^{2} + 2 (x^{2} + 2 x) (2 x + 2) (x^{2})^{2 x}) \cdot 9^{x}

Vereinfache

2 ln (3) \cdot 9^{x} (x^{2})^{2 x} (x^{2} + 2 x)^{2} + (2 x^{4 x} (2 ln (x) + 2) (x^{2} + 2 x)^{2} + 2 (x^{2} + 2 x) (2 x + 2) (x^{2})^{2 x}) \cdot 9^{x} : 2 ln (3) \cdot 9^{x} x^{4 x} (x^{2} + 2 x)^{2} + 9^{x} (4 x^{4 x + 4} ln (x) + 16 x^{4 x + 3} ln (x) + 16 x^{4 x + 2} ln (x) + 4 x^{4 x + 4} + 20 x^{4 x + 3} + 28 x^{4 x + 2} + 8 x^{4 x + 1})

= 2 ln (3) \cdot 9^{x} x^{4 x} (x^{2} + 2 x)^{2} + 9^{x} (4 x^{4 x + 4} ln (x) + 16 x^{4 x + 3} ln (x) + 16 x^{4 x + 2} ln (x) + 4 x^{4 x + 4} + 20 x^{4 x + 3} + 28 x^{4 x + 2} + 8 x^{4 x + 1})

\int \frac{2 x}{( x - 1 ) ( x - 3 )} d x

\int tan^{3} (θ) sec^{4} (θ) d θ

\int (4 - x^{3} - 5 x^{5}) d x

x \to 0 + lim (tan^{x} (6 x))

x \to 0 lim (e^{1})