\( \partial v ( n T \frac { 1 } { 1 - k n } - \frac { v ^ { 2 } } { c ^ { 2 } } + 1 ) \)

It looks like you have a mathematical expression involving a partial derivative, but it's not entirely clear what you're asking for. The expression you provided is: \[ \partial v \left( n T \frac{1}{1 - k n} - \frac{v^2}{c^2} + 1 \right) \] Here, \( \partial v \) typically denotes a partial derivative with respect to the variable \( v \). To compute the partial derivative of the expression inside the parentheses with respect to \( v \), we can differentiate each term separately. Let's denote the expression inside the parentheses as \( f(v) \): \[ f(v) = n T \frac{1}{1 - k n} - \frac{v^2}{c^2} + 1 \] Now, we can compute the partial derivative \( \partial v f(v) \): 1. The first term \( n T \frac{1}{1 - k n} \) is a constant with respect to \( v \), so its derivative is 0. 2. The second term \( -\frac{v^2}{c^2} \) differentiates to \( -\frac{2v}{c^2} \). 3. The third term \( 1 \) is also a constant, so its derivative is 0. Putting it all together, we have: \[ \partial v f(v) = 0 - \frac{2v}{c^2} + 0 = -\frac{2v}{c^2} \] Thus, the result of the partial derivative is: \[ \partial v \left( n T \frac{1}{1 - k n} - \frac{v^2}{c^2} + 1 \right) = -\frac{2v}{c^2} \] If you have a specific context or further questions regarding this expression, please let me know!