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

解

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

2 (x^{3} + 5 x) (3 x^{2} + 5) (5 x - 3)^{4} + 20 (5 x - 3)^{3} (x^{3} + 5 x)^{2}

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} ((5 x - 3)^{4}) = 20 (5 x - 3)^{3}

= 2 (x^{3} + 5 x) (3 x^{2} + 5) (5 x - 3)^{4} + 20 (5 x - 3)^{3} (x^{3} + 5 x)^{2}