PHYSICS –PHY101 VU
© Copyright Virtual University of Pakistan
8
Summary of Lecture 3 – KINEMATICS II
1. The concept of the derivative of a function is exceedingly important. The derivative shows
how fast a function changes when its argument is changed. (Remember that for ( ) we
say that is
fx
fa function that depends upon the argument . You should think of as a
machine that gives you the value when you input .)
2. Functions do not always have to be written as ( ). ( ) is also a
xf
fx
fx xt
0
function. It tells
us where a body is at different times .
3. The derivative of ( ) at time is defined as:
lim
t
t
xt t
dx x
dt t
Δ→
Δ
≡Δ
0
()()
lim .
t
xt t xt
t
Δ→
+Δ −
=
Δ
x
Δ
t
Δ
2
4. Let's see how to calculate the derivative of a simple function like ( ) . We must first
calculate the difference in at two slightly different values, and , while
remembering that
xt t
xttt
=
+Δ
()
()
22
2
22
0
we choose to be extremely small:
2
2 lim 2
t
t
xt t t
ttttt
xxdx
tt
ttdt
Δ→
Δ
Δ= +Δ −
=+Δ +Δ−
ΔΔ
=Δ + ⇒ = =
ΔΔ
1
0
5. In exactly the same way you can show that if ( ) then:
lim
This is an extremely useful result.
n
n
t
xt t
dx x nt
dt t
−
Δ→
=
Δ
≡=
Δ
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