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$f(y)=y^2-1$, where y is non-zero. Differential at y=b is g(u)=2bu This has rank 1 for all b since b is non-zero

$f:R^n->R$ given by $f(x)=x.x$ Then the differential at x is given by u->2x.u. This has rank 1 whenever x is non-zero, since a u can be found such that 2x.u is non-zero e.g. u=x.