derivative of (6x^{2x})

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

(6 x)^{2 x} : 3 6^{x} x^{2 x}

= \frac{d}{d x} (3 6^{x} x^{2 x})

Wende die Produktregel an:

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

f = 3 6^{x}, g = x^{2 x}

= \frac{d}{d x} (3 6^{x}) x^{2 x} + \frac{d}{d x} (x^{2 x}) 3 6^{x}

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

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

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

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

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

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