derivative of (3x^{6x})

Lösung

\frac{d}{d x} ((3 x)^{6 x})

2 ln (3) \cdot 3^{6 x + 1} x^{6 x} + 2 \cdot 3^{1 + 6 x} x^{6 x} (ln (x) + 1)

Schritte zur Lösung

\frac{d}{d x} ((3 x)^{6 x})

Vereinfache

(3 x)^{6 x} : 72 9^{x} x^{6 x}

= \frac{d}{d x} (72 9^{x} x^{6 x})

Wende die Produktregel an:

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

f = 72 9^{x}, g = x^{6 x}

= \frac{d}{d x} (72 9^{x}) x^{6 x} + \frac{d}{d x} (x^{6 x}) 72 9^{x}

\frac{d}{d x} (72 9^{x}) = 2 ln (3) \cdot 3^{6 x + 1}

\frac{d}{d x} (x^{6 x}) = 6 x^{6 x} (ln (x) + 1)

= 2 ln (3) \cdot 3^{6 x + 1} x^{6 x} + 6 x^{6 x} (ln (x) + 1) \cdot 72 9^{x}

6 x^{6 x} (ln (x) + 1) \cdot 72 9^{x} = 2 \cdot 3^{1 + 6 x} x^{6 x} (ln (x) + 1)

= 2 ln (3) \cdot 3^{6 x + 1} x^{6 x} + 2 \cdot 3^{1 + 6 x} x^{6 x} (ln (x) + 1)

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