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

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

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

Wende die Produktregel an:

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

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

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

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

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

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

(x^{2})^{2 x} = x^{4 x}

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