#forward-mode

Forward-mode automatic differentiation evaluates $f(x_1, x_2) = \ln x_1 + x_1 x_2$ at $(x_1, x_2) = (e^2, \pi)$ through the trace $v_1 = \ln x_1$, $v_2 = x_1 x_2$, $f = v_1 + v_2$, carrying a tangent $\dot v$ along with every value. With the seeds $\dot x_1 = 1$ and $\dot x_2 = 0$, what tangent $\dot f$ comes out? Round to 3 decimals.