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 [a,b]
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 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 .

       

When a vector is written in a bold typeface, the magnitude of a vector is often denoted by the same letter in an italic typeface. Thus, we often use r in place of ||r||, and likewise, we often write
v = || v( t) ||
That is, v is the magnitude of v.  
EXAMPLE 2    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
  ( 2 v · a)
=
m v · a

      Check your Reading: What is p' (t) if p(t) = k for all t?