Derivatives of vector functions

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Derivatives and tangent vectors

A derivative of a vector function \(\mathbf{r}:\mathbb{R} \to \mathbb{R}^n\) is defined as

\[\mathbf{r}'(t) = \lim_{h\to 0}\frac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h}\]

The resulting vector is tangent to the curve:

There are more details in the following video:

Try it yourself