derivative f(x)=(2x^2+3)^3(x^2-4)^2

解

導関数

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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