Derivative of a vector-valued function

The tangent vector $\mathbf{r}'(t)$ , found by differentiating each component separately.

The derivative $\mathbf{r}'(t)$ answers the same question a single-variable derivative always has — how is this changing right now — except the answer is a vector, tangent to the curve, pointing in the direction $t$ is increasing. Because a vector-valued limit is taken componentwise, $\mathbf{r}'(t)$ is nothing more than three ordinary derivatives bundled back into a vector. Geometrically, $\mathbf{r}(t+\Delta t) - \mathbf{r}(t)$ is a secant vector connecting two nearby points on the curve, and letting $\Delta t \to 0$ swings that secant into the tangent line.

Examples

For $\mathbf{r}(t) = \langle t^2, t^3, 2t\rangle$ , $\mathbf{r}'(t) = \langle 2t,\ 3t^2,\ 2\rangle$ , so $\mathbf{r}'(1) = \langle 2,3,2\rangle$ is tangent to the curve at $(1,1,2)$ .