derivative of (4x^{5x})

解

\frac{d}{d x} ((4 x)^{5 x})

5 ln (2) \cdot 2^{10 x + 1} x^{5 x} + 5 x^{5 x} (ln (x) + 1) \cdot 102 4^{x}

解答ステップ

\frac{d}{d x} ((4 x)^{5 x})

簡素化

(4 x)^{5 x} : 102 4^{x} x^{5 x}

= \frac{d}{d x} (102 4^{x} x^{5 x})

積の法則を適用:

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

f = 102 4^{x}, g = x^{5 x}

= \frac{d}{d x} (102 4^{x}) x^{5 x} + \frac{d}{d x} (x^{5 x}) 102 4^{x}

\frac{d}{d x} (102 4^{x}) = 5 ln (2) \cdot 2^{10 x + 1}

\frac{d}{d x} (x^{5 x}) = 5 x^{5 x} (ln (x) + 1)

= 5 ln (2) \cdot 2^{10 x + 1} x^{5 x} + 5 x^{5 x} (ln (x) + 1) \cdot 102 4^{x}