Differential of a smooth map
= Differential of a smooth map
{title2=$df_x$}
= Differential
{synonym}
For a <smooth map between manifolds> $f:M\to N$, the <differential> $df_x:T_xM\to T_{f(x)}N$ 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 $d(g\circ f)_x=dg_{f(x)}\circ df_x$.