15. [-11 Points] DETAILS MY NOTES Find \( f^{\prime}(a) \). \[ f(t)=\frac{8 t+50}{t+7} \] \( f^{\prime}(a)=\square \)

To find \( f^{\prime}(a) \), we first need to compute the derivative of the function \( f(t) = \frac{8t + 50}{t + 7} \). We can use the quotient rule, which states that if you have a function in the form of \( \frac{u}{v} \), then the derivative is given by \( \frac{u'v - uv'}{v^2} \), where \( u = 8t + 50 \) and \( v = t + 7 \). Calculating \( u' \) and \( v' \): - \( u' = 8 \) - \( v' = 1 \) Now applying the quotient rule: \[ f'(t) = \frac{(8)(t+7) - (8t + 50)(1)}{(t + 7)^2} \] Simplifying this: \[ f'(t) = \frac{8t + 56 - 8t - 50}{(t + 7)^2} = \frac{6}{(t + 7)^2} \] Finally, to find \( f'(a) \): \[ f'(a) = \frac{6}{(a + 7)^2} \] So, \( f^{\prime}(a) = \frac{6}{(a + 7)^2} \).