Find \( \frac{d s}{d t} \) if \( s=\frac{t}{7 t+8} \)

To find \( \frac{d s}{d t} \) for the function \( s=\frac{t}{7t+8} \), we can use the quotient rule. The quotient rule states that if you have a function \( s = \frac{u}{v} \), then \[ \frac{d s}{d t} = \frac{v \frac{d u}{d t} - u \frac{d v}{d t}}{v^2} \] Here, \( u = t \) and \( v = 7t + 8 \). First, calculate the derivatives: 1. \( \frac{d u}{d t} = 1 \) 2. \( \frac{d v}{d t} = 7 \) Now apply the quotient rule: \[ \frac{d s}{d t} = \frac{(7t + 8)(1) - (t)(7)}{(7t + 8)^2} \] Simplifying this gives: \[ \frac{d s}{d t} = \frac{7t + 8 - 7t}{(7t + 8)^2} = \frac{8}{(7t + 8)^2} \] So, the result is: \[ \frac{d s}{d t} = \frac{8}{(7t + 8)^2} \]