derivative f(x)=x^2sin(x^2+3)

解

導関数

f (x) = x^{2} sin (x^{2} + 3)

2 x sin (x^{2} + 3) + 2 x^{3} cos (x^{2} + 3)

解答ステップ

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

積の法則を適用:

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

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

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

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

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

= 2 x sin (x^{2} + 3) + cos (x^{2} + 3) \cdot 2 x x^{2}

cos (x^{2} + 3) \cdot 2 x x^{2} = 2 x^{3} cos (x^{2} + 3)

= 2 x sin (x^{2} + 3) + 2 x^{3} cos (x^{2} + 3)