In Arbarello–Cornalba–Griffiths–Harris' "*Geometry of algebraic curves*", Vol. 1, Chap. 2, the computation of the tangent cone in the following situation is explained:

Let $f:X\rightarrow Y$ be a morphism between smooth (complex) varieties such that $X$ is birational to $f(X)=Z$. Let $p\in Z$ such that $f^{-1}(p)$ (scheme-theoretic fiber) is smooth. Then the projectivized tangent cone $\mathbb P(\mathcal C_{(Z,Y),p})$ is computed.

Now, let us consider the following situation:

Let $f:X\rightarrow Y$ be a morphism between smooth (complex) varieties such that $X$ is birational to $Z=f(X)$. Assume that there are non-empty subsets $Z_1\subset Z_2\subset Z$ such that the fiber over a point of $Z_2\backslash Z_1$ is two reduced (smooth) points and the fiber over a point of $Z_1$ is one (thus double) point.

Is there a reference where the computation of $\mathbb P(\mathcal C_{(Z,Y),p})$ for $p\in Z_1$ is explained?