4. HIV density The viral load for a particular HIV patient undergoing treatment using ABT-538 is modeled by the function, \[ V=96.39785(0.818656)^{t} \] Where \( V \) is the viral load measured in RNA copies per mL and \( t \) is the time in days since treatment began. Find \( V^{\prime}(1) \) and \( V^{\prime}(10) \) and interpret them.

To find the derivatives \( V'(1) \) and \( V'(10) \), we first need to compute the derivative of the function: \[ V(t) = 96.39785(0.818656)^{t} \] Using the chain rule, the derivative is: \[ V'(t) = 96.39785 \cdot \ln(0.818656) \cdot (0.818656)^{t} \] Now, calculating \( V'(1) \) and \( V'(10) \): 1. **Calculating \( V'(1) \)**: \[ V'(1) = 96.39785 \cdot \ln(0.818656) \cdot (0.818656)^{1} \] \[ \ln(0.818656) \approx -0.199 \] \[ V'(1) = 96.39785 \cdot (-0.199) \cdot 0.818656 \approx -15.286 \] (approximately) 2. **Calculating \( V'(10) \)**: \[ V'(10) = 96.39785 \cdot \ln(0.818656) \cdot (0.818656)^{10} \] \[ (0.818656)^{10} \approx 0.133 \] \[ V'(10) \approx 96.39785 \cdot (-0.199) \cdot 0.133 \approx -2.548 \] (approximately) **Interpretation**: - \( V'(1) \approx -15.286 \): This value indicates that, shortly after treatment begins, the viral load is decreasing rapidly. A negative derivative suggests that the treatment is effective, as the viral load is declining significantly during the first day. - \( V'(10) \approx -2.548 \): By the tenth day, the rate of decline in viral load has slowed compared to the first day. While still negative, the smaller magnitude suggests that although the treatment is still effective, the rate of decrease is less rapid, indicating the viral load is stabilizing. This can inform treatment adjustments as patient responses can vary over time.