(c) By differentiating this consumption function \( C_1 \) (including differentiation of \( W_1 \)) with respect to \( R \), show that

$$ \frac{dC_1}{dR} = -\frac{C_1}{1 + R} + \frac{Y_2}{1 + R} + \frac{y}{1 + R} \left[ -1 - \log(\beta/R) \right]. $$