The derivative of f measures the rate at which the magnitude and direction of f are changing.

The derivative of f is also called the velocity.

The rate of change in velocity is called acceleration.

The second derivative is also called the acceleration.

The idea of taking a limit of a vector-valued function carries over directly from scalar-valued functions.

The derivative of a vector-valued function is a scalar.

Another symbol for f' is

f'(a) is a tangent vector to the path at f(a).

The magnitude of velocity is position.

where f and g are vector-valued functions.