derivative of (2x^3-4x^2*(3x^5+x^2))

解

\frac{d}{d x} ((2 x^{3} - 4 x^{2}) \cdot (3 x^{5} + x^{2}))

48 x^{7} - 84 x^{6} + 10 x^{4} - 16 x^{3}

解答ステップ

\frac{d}{d x} ((2 x^{3} - 4 x^{2}) (3 x^{5} + x^{2}))

積の法則を適用:

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

f = 2 x^{3} - 4 x^{2}, g = 3 x^{5} + x^{2}

= \frac{d}{d x} (2 x^{3} - 4 x^{2}) (3 x^{5} + x^{2}) + \frac{d}{d x} (3 x^{5} + x^{2}) (2 x^{3} - 4 x^{2})

\frac{d}{d x} (2 x^{3} - 4 x^{2}) = 6 x^{2} - 8 x

\frac{d}{d x} (3 x^{5} + x^{2}) = 15 x^{4} + 2 x

= (6 x^{2} - 8 x) (3 x^{5} + x^{2}) + (15 x^{4} + 2 x) (2 x^{3} - 4 x^{2})

簡素化

(6 x^{2} - 8 x) (3 x^{5} + x^{2}) + (15 x^{4} + 2 x) (2 x^{3} - 4 x^{2}) : 48 x^{7} - 84 x^{6} + 10 x^{4} - 16 x^{3}

= 48 x^{7} - 84 x^{6} + 10 x^{4} - 16 x^{3}