Example 8.40 (Introduction to smooth manifolds. Lee)
If we consider $\mathbb{R}^n$ as a Lie group under addition, left translation by an element $b\in\mathbb{R}^n$ is given by the affine map $L_b(x)=b+x$, whose differential $d(L_b)$ is represented by the identity matrix in standard coordinates. Thus a vector field $X^i \partial_{x_i}$ is left-invariant if and only if its coefficients $X^i$ are constants.
Question 1. Why $X^i$ are constants?
My attempt: $X^i(x)\left.\partial_{x_i}\right|_{x}=X^i(b+x)\left.\partial_{x_i}\right|_{b+x}$