Why does $L_a(f(b)) =0$ in Proposition $7.8$ of Clader and Ross Beginning in Algebraic Geometry

I have been self studying Algebraic Geometry from Clader and Ross's Beginning in Algebraic Geometry and I have a question on Page $195$: Proposition $7.8$.

Background information: If $f \in K[x_1,...,x_n]$ and $a=(a_1,...,a_n)\in \mathbb{A}^n$ , then the linearization of $f$ at $a$ is defined by $L_a(f) = f(a) +\sum_{i=1}^n (\frac{\partial f}{\partial{x_i}}(a)(x_i-a_i)) \in K[x_1,...,x_n]$.

Let $X\in \mathbb{A}^n$ be an affine variety and $a\in X$ . The linearlization of $X$ at $a$ is $L_aX= V(${$L_af| f\in I(X)$}$) \subseteq \mathbb{A}^n$.

Let $X \subseteq \mathbb{A}^n$ be an affine variety and $a=(a_1,...,a_n)\in X$ . For any $b=(b_1,....,b_n)\in L_aX$, the tangent vector associated to $b$ is defined by ${ab}^{\rightarrow}=(b_1-a_1,...,b_n-a_n) \in K^n$. The tangent space of $X$ at $a$ is the collection of tangent vectors: $T_aX = ${${ab}^{\rightarrow}|b \in L_aX$}$\subseteq K^n$.

Proposition $7.8$ Let $X\subseteq \mathbb{A}^n$ be an affine variety and let $a\in X$. Then $T_aX=${$ v\in K^n| \nabla f(a).v =0$ for all $ f\in I(X))$}.

Proof:Let $v = (v_1,...,v_n) \in K^n$ $v\in T_aX $ iff $v = {ab}^{\rightarrow}$ for some $b\in LaX$ which is equivalent to the requirement that $b=(v_1+a_1,...,v_n+a_n)\in L_a X$

By the definition of $L_a X$, we have that $b \in L_aX$ iff for all $f \in I(X)$, we have $0 = L_af(b) = \sum_{i=1}^n [\frac{\partial f}{\partial {x_j}}(a)](b_i-a_i)= \nabla {f(a)} .v$. ...

I am not able to understand how $L_a f(b)=0$ in the previous step and how the constant term $f(a)$ in the right hand side of the equality of $L_a f(b)= \sum_{i=1}^n [\frac{\partial f}{\partial {x_j}}(a)](b_i-a_i)$ equals $0$?

Please help me with these $2$ questions.

Thank you very much