derivative of (2x^{6x})

Lösung

\frac{d}{d x} ((2 x)^{6 x})

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

Schritte zur Lösung

\frac{d}{d x} ((2 x)^{6 x})

Vereinfache

(2 x)^{6 x} : 6 4^{x} x^{6 x}

= \frac{d}{d x} (6 4^{x} x^{6 x})

Wende die Produktregel an:

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

f = 6 4^{x}, g = x^{6 x}

= \frac{d}{d x} (6 4^{x}) x^{6 x} + \frac{d}{d x} (x^{6 x}) 6 4^{x}

\frac{d}{d x} (6 4^{x}) = 3 ln (2) \cdot 2^{6 x + 1}

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

= 3 ln (2) \cdot 2^{6 x + 1} x^{6 x} + 6 x^{6 x} (ln (x) + 1) \cdot 6 4^{x}

6 x^{6 x} (ln (x) + 1) \cdot 6 4^{x} = 3 \cdot 2^{1 + 6 x} x^{6 x} (ln (x) + 1)

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

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