Topic 2.1 Extended
D – The derivative
Recall that the instantaneous velocity is given by
v =
limit
t→0
x
t
Instantaneous
Velocity
We can now express the instantaneous velocity in
function notation:
v =
limit
t→0
x(t + t) - x(t)
t
Instantaneous
Velocity
function notation
Topic 2.1 Extended
D – The derivative
v =
limit
t→0
x(t + t) - x(t)
t
Instantaneous
Velocity
function notation
Graphically, as t→0, the secant line becomes the
tangent line.
x
secant
secant
secant
line
line
secant
line
tangent
line
line
x(t)
x(t
x(t
x(t
x(t
+ t)
+ t)
+ t)
+ t)
x(t)
t + t
t + t
t + t
t + t
t
t
Topic 2.1 Extended
D – The derivative
v =
limit
t→0
x(t + t) - x(t)
t
Instantaneous
Velocity
function notation
The process of obtaining a formula for v when given
a formula for x consists of the following
4 steps:
STEP 1: Find and simplify x(t + t).
STEP 2: Subtract x(t) - the original function - from STEP 1's result.
STEP 3: Divide the difference from STEP 2 by t.
STEP 4: Let all the remaining t's become zero.
FYI: We will only have to go through this process for a few days, after
which we will have discovered an excellent shortcut.
FYI: This 4-step process is called "taking the derivative."
Topic 2.1 Extended
D – The derivative
v =
limit
t→0
x(t + t) - x(t)
t
Instantaneous
Velocity
function notation
Consider our old standby function, x(t) = 2t2.
We will now "take the derivative" of it using the
four step process:
STEP 1: Find and simplify x(t + t).
Since x(t) = 2t2,
x(t + t) = 2(t + t)2 = 2(t2 + 2tt + t2)
x(t + t) = 2t2 + 4tt + 2t2
STEP 2: Subtract x(t) - the original function - from STEP 1's result.
x(t + t) - x(t) = 2t2 + 4tt + 2t2 - 2t2
x(t + t) - x(t) = 4tt + 2t2
Topic 2.1 Extended
D – The derivative
v =
limit
t→0
x(t + t) - x(t)
t
Instantaneous
Velocity
function notation
Consider our old standby function, x(t) = 2t2.
We will now "take the derivative" of it using the
four step process:
STEP 3: Divide the difference from STEP 2 by t.
x(t + t) - x(t)
4tt + 2t2
=
t
t
x(t + t) - x(t)
= 4t + 2t
t
STEP 4: Let all the remaining t's become zero.
0
v = 4t
The derivative of 2t2
4t + 2t