Final answer:
To find the derivative of a vector-valued function, take the derivative component by component and with respect to time. The time derivative of the position function gives us the velocity function, and the derivative of the velocity function gives us the acceleration. The acceleration can also be expressed in terms of the second derivative of the position function.
Step-by-step explanation:
To find the derivative of a vector-valued function, we take the derivative component by component. When finding the derivative of the velocity function to get the acceleration, we take the first derivative with respect to time. Similarly, the time derivative of the position function gives us the velocity function. The acceleration can also be expressed in terms of the second derivative of the position function. Taking derivatives with respect to time helps us find the acceleration of a given function.