 Differential of a smooth map

For a smooth map between manifolds $f:M\to N$, the differential $d{f}_x:T_{x}M\to T_{f(x)}N$ is the linear map obtained by differentiating in charts. Intrinsically it sends the tangent vector of a curve $\gamma$ through $x$ to the tangent vector of $f\circ \gamma$. The chain rule makes it independent of charts and gives $d(g\circ f{)}_x=d{g}_{f(x)}\circ d{f}_x$.