Let $(M,\mathcal A_M)$ and $(N,\mathcal A_N)$ be $2$ smooth manifolds and $f\in\mathcal C^\infty(M,N)$. For all $x\in M$, define $\mathcal C_x^\infty(M)$ the set of all smooth paths $c$ defined on a neighborhood of $0\in\mathbb R$ and such that $c(0)=x$. Then, we define the equivalence relation $\sim$ on $\mathcal C_x^\infty(M)$ by : $$c_1\sim c_2\iff\text{there is a chart $\varphi$ in $x$, $(\varphi\circ c_1)'(0)=(\varphi\circ c_2)'(0)$}$$ From there, the differential of $f$ in $x\in M$ is defined as the map induced by the quotient of the map $c\in\mathcal C_X^\infty(M)\longmapsto f\circ c$ by $\sim$ and is noted $df_x$
Using this definition, I can find coherent results such as the fact that the differential of a constant function is $0$. Indeed, for $f\equiv a$ we have : $$df_x(\overline c)=(f\circ c)'(0)=0$$ where $\overline c=\mathrm{cl}_\sim(c)$
But this seems quite difficult to use in practice. My question is : how do proceed to differentiate functions that are slightly more complex than simply constant functions?
Say for example if we define $f$ as : $$f:(x_1,...,x_n)\in\mathbb R^n/\mathbb Z^n\longmapsto\left(\cos(2\pi x_1),\sin(2\pi x_1),...,\cos(2\pi x_n),\sin(2\pi x_n)\right)\in\left(\mathbb S^1\right)^n=\mathbb T^n$$ then for $x\in\mathbb R^n/\mathbb Z^n$, how does one calculate $df_x$?