A function $u$ of two variables is defined implicitly by $u(x,t)=f(x-tu(x,t))$, where f is a given bounded, differentiable function of one variable, $f : R → R$.
Q:Given that $f(s)=1−\tanh s$,find the smallest positive number $t_m$ such that $\cfrac{∂u}{∂x}$ is undefined for precisely one value of $x$.
I found that $\cfrac{∂u}{∂x} = \cfrac{ f′(x−tu(x,t))}{1+tf′(x−tu(x,t))}$ but have no idea on what to do next.