derivative of 2x{z}(xcos(sin(x)))

解

\frac{d}{d x} (2 x z (x) cos (sin (x)))

2 (cos (sin (x)) z (x) + x (cos (sin (x)) \frac{d}{d x} (z (x)) - sin (sin (x)) cos (x) z (x)))

解答ステップ

\frac{d}{d x} (2 x z (x) cos (sin (x)))

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 \frac{d}{d x} (x z (x) cos (sin (x)))

積の法則を適用:

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

f = x, g = z (x) cos (sin (x))

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

\frac{d x}{d x} = 1

\frac{d}{d x} (z (x) cos (sin (x))) = cos (sin (x)) \frac{d}{d x} (z (x)) - sin (sin (x)) cos (x) z (x)

= 2 (1 \cdot z (x) cos (sin (x)) + (cos (sin (x)) \frac{d}{d x} (z (x)) - sin (sin (x)) cos (x) z (x)) x)

簡素化

= 2 (cos (sin (x)) z (x) + x (cos (sin (x)) \frac{d}{d x} (z (x)) - sin (sin (x)) cos (x) z (x)))