Part 1: Properties of the Derivative

In the previous section, we showed that the velocity v( t) of a vector-valued function r( t) is tangent to the curve parameterized by r( t) at "time" t. In this section, we explore additional properties of the derivative of a vector-valued function.

To begin with, the following notations are commonly used to denote derivatives of vector values functions:

r' ( t) =    d dt r( t) =    dr dt  =   × r   ( t)

In particular, a dotted r is used if t is interpreted to be "time", while r' ( t) is used if no interpretation of t is to be inferred. Also, the operator notation allows us to state the following theorem.       

Theorem 7.1: If p( t) and q(t) are differentiable vector-valued functions over an interval I, if f( t) is differentiable for all t, and if a,b are scalars, then the following hold over that interval.

1.  d dt [ ap( t) +bq(t) ] = a  dp dt +b  dq dt 2.  d dt [ f( t) p( t) ] = f' ( t) p( t) +  f( t) p' ( t) 3.  d dt p( f( t) ) = p' ( f( t) )    d dt f( t)  when f( t)  is in I 4.  d dt [ p( t) ·q(t) ] = p' ( t) ·q(t) + p( t) ·q' (t) 5.  d dt [ p( t) ×q(t) ] = p' ( t) ×q( t) + p( t) ×q' (t)

These properties follow directly from the definition of r' ( t) , and they are used frequently in elementary mechanics, as the next two examples illustrate.

EXAMPLE 1    The kinetic energy of an object with a constant mass m and position r( t) at time t is defined to be

K =  1 2 mv2

where v² = v·v  and v = r' ( t).  What is K' ( t) ?    

Solution:  Since K = m v·v / 2, property 4 of theorem 7.1 implies that

K' ( t) =  m 2  d dt ( v·v) =  m 2 é ë æ è  d dt v ö ø ·v+v· æ è  d dt v ö ø ù û =  m 2 ( a·v+v·a) =  m 2 ( 2v·a) = m v·a

EXAMPLE 2    The angular velocity of a vector-valued function r( t) is defined to be

L( t) = r( t) ×v(t)

where v( t) = r' ( t) and a = r'' ( t) . Use property 5 in theorem 7.1 to find L' ( t) .    

Solution:  Property 5 implies that

L' ( t) = æ è  d dt r(t) ö ø ×v+r( t) ×  d dt v = v×v+r×a

Thus, L' ( t) = r×a .