derivative of sin^2(2xcos^3(3x))

解

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

2 cos^{3} (3 x) sin (4 x) - 9 cos^{2} (3 x) sin^{2} (2 x) sin (3 x)

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (cos^{3} (3 x)) = - 9 cos^{2} (3 x) sin (3 x)

= 2 sin (4 x) cos^{3} (3 x) + (- 9 cos^{2} (3 x) sin (3 x)) sin^{2} (2 x)

簡素化

= 2 cos^{3} (3 x) sin (4 x) - 9 cos^{2} (3 x) sin^{2} (2 x) sin (3 x)