Differentiate. \[ y=\frac{2 x}{7-\tan (x)} \]

To differentiate the function \( y = \frac{2x}{7 - \tan(x)} \), we will use the quotient rule. The quotient rule states that if \( y = \frac{u}{v} \), then \[ y' = \frac{u'v - uv'}{v^2} \] where \( u = 2x \) and \( v = 7 - \tan(x) \). First, we find the derivatives of \( u \) and \( v \): 1. \( u' = 2 \) 2. To differentiate \( v \), note that \( v' = -\sec^2(x) \) because the derivative of \( \tan(x) \) is \( \sec^2(x) \). Now, substituting into the quotient rule: \[ y' = \frac{(2)(7 - \tan(x)) - (2x)(-\sec^2(x))}{(7 - \tan(x))^2} \] This simplifies to: \[ y' = \frac{14 - 2\tan(x) + 2x\sec^2(x)}{(7 - \tan(x))^2} \] Thus, the derivative of \( y \) is: \[ y' = \frac{14 - 2\tan(x) + 2x\sec^2(x)}{(7 - \tan(x))^2} \]