This is an exercise left for homework, taken from the textbook which we follow in class, Lee's book ''Introduction to Smooth Manifolds'' Second Edition.
Definition: $D_vW(p)=\frac{\mathrm{d} }{\mathrm{d} t}|_{t=0} W_{p+tv}=\lim_{t\rightarrow 0}\frac{1}{t}(W_{p+tv}-W_p)$ (*)
Suppose $v\in \mathbb{R}^n$ and $W$ is a smooth vector field on an open subset of $\mathbb{R}^n$. Show that the directional derivative $D_vW(p)$ defined by (*) is equal to $(L_VW)_p$, where $V$ is the vector field $V=v^i\frac{\partial }{\partial x^i}$ with constant coefficients in standard coordinates.
Any thoughts? Am i supposed to use the theorem which states $L_VW=[V,W]$, or is there a simpler way to show this equality? Thanks in advance for your answers!
Consider the function defined by $f(t)=x+tv$, $W(x+tv)=W(f(x)), {d\over{dt}}W(x+tv)={d\over{dt}}W(f(x))=dW({d\over{dt}}f(t))=dW(V(x))$.
This the directional derivative, but not the Lie derivative.
https://en.m.wikipedia.org/wiki/Lie_derivative