derivative of (6x^{6x})

解

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

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

解答ステップ

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

簡素化

(6 x)^{6 x} : 4665 6^{x} x^{6 x}

= \frac{d}{d x} (4665 6^{x} x^{6 x})

積の法則を適用:

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

f = 4665 6^{x}, g = x^{6 x}

= \frac{d}{d x} (4665 6^{x}) x^{6 x} + \frac{d}{d x} (x^{6 x}) 4665 6^{x}

\frac{d}{d x} (4665 6^{x}) = ln (6) \cdot 6^{6 x + 1}

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

= ln (6) \cdot 6^{6 x + 1} x^{6 x} + 6 x^{6 x} (ln (x) + 1) \cdot 4665 6^{x}

6 x^{6 x} (ln (x) + 1) \cdot 4665 6^{x} = 6^{1 + 6 x} x^{6 x} (ln (x) + 1)

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