Part 2/7:

Just as with real-valued functions, calculating the second derivative of a vector function involves taking derivatives repeatedly. Let’s consider a vector function (\mathbf{r}(t)), where (t) typically represents time. The first derivative, (\mathbf{r}'(t)), signifies the velocity vector, while the second derivative, (\mathbf{r}''(t)), represents acceleration.

Practical Example

Suppose for a specific vector function (\mathbf{r}(t)), the first derivative (\mathbf{r}'(t)) is given by:

[

\mathbf{r}'(t) = \left( -2 \sin t, \cos t, 1 \right)

]

To find the second derivative (\mathbf{r}''(t)), differentiate each component again:

• Derivative of (-2 \sin t) is (-2 \cos t).

• Derivative of (\cos t) is (- \sin t).

• Derivative of (1) is (0).