derivative of sin(2^{sqrt(1+1/x)})

解答

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

- \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}} cos ( 2 ^{\frac{x + 1}{x}} )}{x ^{\frac{3}{2}} x + 1}

求解步骤

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

使用连锁律:

cos (2^{1 + \frac{1}{x}}) \frac{d}{d x} (2^{1 + \frac{1}{x}})

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

\frac{d}{d x} (2^{1 + \frac{1}{x}}) = - \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}}}{x ^{\frac{3}{2}} x + 1}

= cos (2^{1 + \frac{1}{x}}) - \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}}}{x ^{\frac{3}{2}} x + 1}

化简

cos (2^{1 + \frac{1}{x}}) - \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}}}{x ^{\frac{3}{2}} x + 1} : - \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}} cos ( 2 ^{\frac{x + 1}{x}} )}{x ^{\frac{3}{2}} x + 1}

= - \frac{ln ( 2 ) \cdot 2 ^{\frac{x + 1 - x}{x}} cos ( 2 ^{\frac{x + 1}{x}} )}{x ^{\frac{3}{2}} x + 1}