derivative f(x)=(2x^3+3x)(x-3)(x+5)

解

導関数

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

10 x^{4} + 16 x^{3} - 81 x^{2} + 12 x - 45

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

(6 x^{2} + 3) (x - 3) (x + 5) + (2 x + 2) (2 x^{3} + 3 x) : 10 x^{4} + 16 x^{3} - 81 x^{2} + 12 x - 45

= 10 x^{4} + 16 x^{3} - 81 x^{2} + 12 x - 45