derivative of (e^{x^2}+4^2*(4x)^{3x})

解

\frac{d}{d x} ((e^{x^{2}} + 4)^{2} \cdot (4 x)^{3 x})

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

解答ステップ

\frac{d}{d x} ((e^{x^{2}} + 4)^{2} (4 x)^{3 x})

簡素化

(e^{x^{2}} + 4)^{2} (4 x)^{3 x} : 6 4^{x} x^{3 x} (e^{x^{2}} + 4)^{2}

= \frac{d}{d x} (6 4^{x} x^{3 x} (e^{x^{2}} + 4)^{2})

積の法則を適用:

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

f = 6 4^{x}, g = x^{3 x} (e^{x^{2}} + 4)^{2}

= \frac{d}{d x} (6 4^{x}) x^{3 x} (e^{x^{2}} + 4)^{2} + \frac{d}{d x} (x^{3 x} (e^{x^{2}} + 4)^{2}) 6 4^{x}

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

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

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