((4x^2+5)^2*(7x)^{2x})^'

解

((4 x^{2} + 5)^{2} \cdot (7 x)^{2 x})^{^{'}}

2 ln (7) \cdot 4 9^{x} x^{2 x} (4 x^{2} + 5)^{2} + (2 x^{2 x} (ln (x) + 1) (4 x^{2} + 5)^{2} + 16 x^{1 + 2 x} (4 x^{2} + 5)) \cdot 4 9^{x}

解答ステップ

((4 x^{2} + 5)^{2} (7 x)^{2 x})^{'}

簡素化

(4 x^{2} + 5)^{2} (7 x)^{2 x} : 4 9^{x} x^{2 x} (4 x^{2} + 5)^{2}

= (4 9^{x} x^{2 x} (4 x^{2} + 5)^{2})^{^{'}}

積の法則を適用:

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

f = 4 9^{x}, g = x^{2 x} (4 x^{2} + 5)^{2}

= (4 9^{x})^{^{'}} x^{2 x} (4 x^{2} + 5)^{2} + (x^{2 x} (4 x^{2} + 5)^{2})^{^{'}} 4 9^{x}

(4 9^{x})^{'} = 2 ln (7) \cdot 4 9^{x}

(x^{2 x} (4 x^{2} + 5)^{2})^{'} = 2 x^{2 x} (ln (x) + 1) (4 x^{2} + 5)^{2} + 16 x^{1 + 2 x} (4 x^{2} + 5)

= 2 ln (7) \cdot 4 9^{x} x^{2 x} (4 x^{2} + 5)^{2} + (2 x^{2 x} (ln (x) + 1) (4 x^{2} + 5)^{2} + 16 x^{1 + 2 x} (4 x^{2} + 5)) \cdot 4 9^{x}