Let $f:M\rightarrow N$ be a smooth map between smooth manifolds, let $p\in M$ and $v\in T_{p}M$. Two different definitions of differential maps on tangent space: let $\gamma$ be a smooth curve on $M$ representing $v$ ($\gamma(0)=p,\gamma'(0)=v)$ and define df_{p}(v)=(f\circ v)'(0).
Second definition: let $\mathcal{D}$ be a derivation at $p$, $g:N\rightarrow\mathbb{R}$ be a smooth function, then define $df_{p}(\mathcal{D})(g)=\mathcal{D}(g\circ f)$. We want to show that the two definitions of $df_{p}$ coincide.
Consider the matrix representation of the second definition of $df_{p}$ in local coordinates, this is: $\left[ df_{p}=\frac{\partial\hat{f}^{j}}{\partial x^{i}}\right]$
where $\hat{f}$ is the coordinate representation of $f$.
In the first definition, let $x_{1},\dots,x_{n}$ be local coordinates at $p$. Then, $\gamma(t)=(\gamma_{1}(t),\dots,\gamma_{n}(t))$ where $\gamma_{i}(t)=x_{i}(\gamma(t))$. The matrix representation of $df_{p}$ is:$\left[ df_{p}=\sum_{i=1}^{n}\frac{\partial f}{\partial x^{i}}\right]$
I might have gotten the matrix representation of $df_{p}$ in the first definition incorrectly, but of course if these matrices are the same with respect to these coordinates, this means that the two definitions are coincide. I am wondering are the steps I have done right.