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

解

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

21 x^{6} + 12 x^{5} - 5 x^{4}

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (3 x^{4} - x^{3}) = 12 x^{3} - 3 x^{2}

= (2 x + 3 x^{2}) (3 x^{4} - x^{3}) + (12 x^{3} - 3 x^{2}) (x^{2} + x^{3})

簡素化

(2 x + 3 x^{2}) (3 x^{4} - x^{3}) + (12 x^{3} - 3 x^{2}) (x^{2} + x^{3}) : 21 x^{6} + 12 x^{5} - 5 x^{4}

= 21 x^{6} + 12 x^{5} - 5 x^{4}