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

We are given \[ f(t)=\frac{8t+50}{t+7}. \] To find \( f^{\prime}(a) \), we first compute \( f^{\prime}(t) \) using the quotient rule. The quotient rule states that if \[ f(t)=\frac{g(t)}{h(t)}, \] then \[ f^{\prime}(t)=\frac{g^{\prime}(t)h(t)-g(t)h^{\prime}(t)}{[h(t)]^2}. \] **Step 1. Identify \( g(t) \) and \( h(t) \):** \[ g(t)=8t+50 \quad \text{and} \quad h(t)=t+7. \] **Step 2. Compute the derivatives:** \[ g^{\prime}(t)=8 \quad \text{and} \quad h^{\prime}(t)=1. \] **Step 3. Apply the quotient rule:** \[ f^{\prime}(t)=\frac{8(t+7)-(8t+50)(1)}{(t+7)^2}. \] **Step 4. Simplify the numerator:** \[ 8(t+7)=8t+56, \] so the numerator becomes \[ 8t+56-(8t+50)=8t+56-8t-50=6. \] **Step 5. Write the final derivative:** \[ f^{\prime}(t)=\frac{6}{(t+7)^2}. \] Finally, replacing \( t \) with \( a \), we obtain \[ f^{\prime}(a)=\frac{6}{(a+7)^2}. \]