In this lecture we define the derivative of a smooth map $\varphi \colon M \to N$ between two manifolds. At each point $x \in M$, the derivative is a linear map

$$D \varphi(x) \colon T_x M \to T_{ \varphi(x)}N.$$

We then define the tangent bundle $T M$ as the disjoint union of all the tangent spaces:

$$T M := \bigsqcup_{x \in M} T_xM.$$

and prove this is naturally a smooth manifold of twice the dimension of $M$. Finally, the maps $D \varphi(x)$ combine to give a single smooth map $D \varphi \colon T M \to TN$.

Along the way we give an alternative definition of tangent vectors as velocity vectors $\gamma’(t)$ of curves $\gamma(t)$. This approach is often more useful in computations than our earlier definition of tangent vectors as derivations.