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Unit 2 – THE DERIVATIVE, SLOPE AND RATE OF CHANGE
General Objective of the Unit:
At the end of the unit the student should be able to comprehend with the derivative, slope and rate of change.
Specific Objectives:
At the end of the unit, the student is expected to:
- Familiar with the derivative of the different functions;
- Determine the derivative of the given functions;
- Determine the slope of the given functions; and
- Find the rate of change of the given function.
Content:
Learning Activity 2.1: Familiarization of the derivative
In the function y f x , when x changes to the value x x , there is also
a corresponding change in y to the value y y. We call x an increment of x
and y an increment in y.
Hence, we can write
y f x x f x
y y f x x
y f x
In Figure 2.1, the ratio x
y
is the slope of the line that joint the points
P x , y and P ' x x , y y . Moreover, as x approaches zero P ' approaches
P along the curve and in ordinary cases the line PP ' approaches a certain straight line ( PT in the figure) as a limiting position.
Therefore, the ratio x
y
approaches a limit = the slope of the line PT.
This limit is called the derivative of y with respect to x.
Fundamental Definition: The derivative of y with respect to x is the limit
of the ratio x
y
when x approaches zero.
The derivative is designed by the symbol dx
dy
. Thus, we have
x
f x x f x
Lim
x
y
Lim
dx
dy
x x
0 0
Other symbols for the derivatives are:
y ' ; f ' x ; Dxy ; f x
dx
d
Learning Activity 2.2: Determining the derivative
From the function,
y f x
y y f x x
By subtraction, we get
y f ^ x x ^ f x
Dividing through by x , we obtained
x
f x x f x
x
y
Determine the limit of x
y
as x approaches zero.
P
P '
T
x
y y f x
Fig. 2..
y
y
x
x x x x
y Lim
x
y
Lim
x x x
x
x x x
x x x
y x x x
y x x x
y y x x
y x
x x 2
0 0
Example 4: differentiate y^ sin x
Solution:
y x x x x x x x x x
y y x x x x x x
y x
sin cos cos sin sin cos sin sin 1 cos
sin sin cos cos sin
sin
Since sin 2 A 21 1 cos 2 A , then, we can have 1 cos x 2 sin^221 x , so we may
write
y x
x
x
x
x
x
x
x
x
y
y x x x x
' cos
sin
sin
2 sin
sin
cos
sin cos 2 sin sin
2
1
2
1
2
1
2
1
2
(^21)
The students should also recall other trigonometric formula for the sum and difference of two angles.
Example 5: Differentiate the function y^ tan x
Solution:
y x
x x
x
x
x
x
y
x x
x x
x
x x
x x
y
x x
x x
y y x x
y x
2
2
2
' sec
1 tan tan
sec
tan
1 tan tan
tan 1 tan
tan
1 tan tan
tan tan
1 tan tan
tan tan
tan
tan
Learning Activity 2.3: Determining the slope of a function
The tangent to the plane curve is a straight that intersects a curve in two or more distinct points is called a secant.
Let P be a fixed point on the curve near point P '. In Figure 2.2, the segment PT is tangent to the curve at point P. If we let P ' be made to approach P along the curve, the secant PP ' approaches, in general, a limiting position. Then, the straight line which is that limit, PT in Fig. 2.2 is called the tangent to the curve at P.
The slope of the tangent to the curve at any point is called the slope of the curve at that point. That is, when P ' approaches P , the slope of the secant approaches as its limit the slope of the curve.
The slope of the secant PP ' is x
y
. When x approaches zero P '
approaches P along the curve so that the slope of the secant approaches at its limit the slope of the curve at P.
Hence, we can say that:
- The derivative of a function is identical with the slope of the graph of the function
Fig. 2..
dt t t
dr
t t t t
r
t t t
t
r
r r t t
t
r
t r
S r
2
2
^
When t ^2 ,
- 2 2 2
1 dt
dr in /sec
Example 2: The radius of a sphere, initially zero increase at the rate of 6 ft/sec. Find how fast the volume is increasing after 14 second.
Solution: Let V the volume of the sphere ft^3 /sec; (^) r the radius of the
sphere ft and t time sec . Since r 6 t , we have
54 /sec
3
4 2
2
3 3
(^34) 3
(^34) 3
4
t ft
dt
dV
t
dt
dV
V r t t
References:
- Love and Rainville. Differential and Integral Calculus.
- Lethold, et al. Calculus with Analytic Geometry.
- Peterson, Thurman S. Calculus with Analytic Geometry.
