derivative of (2x^{2x})

Lösung

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

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

Schritte zur Lösung

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

Vereinfache

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

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

Wende die Produktregel an:

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

f = 4^{x}, g = x^{2 x}

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

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

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

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

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

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

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