The derivative of vector-valued function r with respect to parameter
t, where 
, is given by
provided that the component functions are differentiable functions. One might choose to understand as the tangent vector to the space curve of the motion at time t. The second derivative is obtained in the obvious way, by differentiating the derivative function component by component.
Speed of the point in motion is given by the norm of the vector derivative:
Smooth: a space curve given by the vector function r(t) on an
interval I is called smooth if r' is continuous and 
on I, with the possible exception of the end points.
One operatives similarly for integrals:
where R is an anti-derivative of r. The usual integration rules apply....
The usual rules of differentiation apply: suppose that u and v are differentiable vector-valued functions, c is a scalar, and f is a real-valued function. Then
None to speak of.
The (very good!) news is that the basic operations of differentiation and integration for vector-valued functions are carried out exactly the way we would expect, including the chain rule, the sum rule, and three different versions of the product rule!