derivative of log_{2}(sin(2x+x))

解答

\frac{d}{d x} (lo g_{2} (sin (2 x) + x))

\frac{2 cos ( 2 x ) + 1}{ln ( 2 ) ( sin ( 2 x ) + x )}

求解步骤

\frac{d}{d x} (lo g_{2} (sin (2 x) + x))

使用对数计算法则:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

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

将常数提出:

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

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

使用连锁律:

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

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

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

= \frac{1}{ln ( 2 )} \cdot \frac{1}{sin ( 2 x ) + x} (cos (2 x) \cdot 2 + 1)

化简

\frac{1}{ln ( 2 )} \cdot \frac{1}{sin ( 2 x ) + x} (cos (2 x) \cdot 2 + 1) : \frac{2 cos ( 2 x ) + 1}{ln ( 2 ) ( sin ( 2 x ) + x )}

= \frac{2 cos ( 2 x ) + 1}{ln ( 2 ) ( sin ( 2 x ) + x )}