derivative of 1/2 tan(xsin(2x))

解

\frac{d}{d x} (\frac{1}{2} tan (x) sin (2 x))

sin (2 x)

解答ステップ

\frac{d}{d x} (\frac{1}{2} tan (x) sin (2 x))

定数を除去:

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

= \frac{1}{2} \frac{d}{d x} (tan (x) sin (2 x))

積の法則を適用:

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

f = tan (x), g = sin (2 x)

= \frac{1}{2} (\frac{d}{d x} (tan (x)) sin (2 x) + \frac{d}{d x} (sin (2 x)) tan (x))

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

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

= \frac{1}{2} (sec^{2} (x) sin (2 x) + cos (2 x) \cdot 2 tan (x))

簡素化

\frac{1}{2} (sec^{2} (x) sin (2 x) + cos (2 x) \cdot 2 tan (x)) : sin (2 x)

= sin (2 x)