$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes time, could any one tell me how to find implicit differentiation of the relation $y_c(\tau,v_c;t)=0$ and from there how we get $\frac{\partial}{\partial t} v_c(\tau;t)|_{t=0}=-\frac{1}{2} a_c(\tau)$ where $a_c(\tau)$ denote the initial acceleration.
page $2726,$ The paper from which I reading this
Chillingworth, D. R. J. "Dynamics of an impact oscillator near a degenerate graze." Nonlinearity 23.11 (2010): 2723.
