Answer
$0.02$ ohm/sec
Work Step by Step
Step 1. Identify the given conditions: $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$, $\frac{dR_1}{dt}=-1$ ohm/sec, $\frac{dR_2}{dt}=0.5$ ohm/sec, $R_1=75$ ohms, $R_2=50$ ohms,
Step 2. Calculate $R$: $\frac{1}{R}=\frac{1}{75}+\frac{1}{50}=\frac{50+75}{50\times75}=\frac{125}{50\times75}=\frac{1}{30}$, thus $R=30$ ohms
Step 3. Take derivatives: $-\frac{1}{R^2}\frac{dR}{dt}=-\frac{1}{R_1^2}\frac{dR_1}{dt}-\frac{1}{R_2^2}\frac{dR_2}{dt}$
Step 4. Thus, $\frac{dR}{dt}=R^2(\frac{1}{R_1^2}\frac{dR_1}{dt}+\frac{1}{R_2^2}\frac{dR_2}{dt})=30^2(\frac{1}{75^2}(-1)+\frac{1}{50^2}(0.5))=0.02$ ohm/sec
