1. By :
Bagus Mustiko Aji (133194047 )
Ia’natusyarifah (133194049)

2. Direct use of the definition
Is generally not the most efficient way to find the derivative of a function.
Except for very simple functions, the definition does not produce the
derivative quickly and easily. However, the definition can be used to prove
general rules that enable you to find the derivatives of most functions in a
reasonable length of time.

3. Before studiying the power rule and the sum rule, we should out that f ’(x) is
not the only way to represent the derivative. Notation that are frequently used
include:
Each function that appears in the rules of this chapter is assumed to be
differentiable.

4. 
The graph in figure 1 of the function f(x) = c is a horizontal line
whose slope equals zero for all values of x. The rule for the
constant function can be obtained from the definition.

5. The derivatives listed in Table 1 were found for some simple power functions.
Function
Derivativ
e
f(x) = x2
f’(x) = 2x
f(x) = x3
f’(x) = 3x2
f(x) = x4
f’(x) = 4x3
These results are examples of the power rule. It is used for functions of the
type f(x) = xn, where n is a real number.

6. 

8. 

10. 

12. The interpretation of the derivative as the intantaneous rate of change of the dependent
variable with respect to the independent variable is emphasized.
A. Velocity
The average velocity of a moving object between t=t1 and t=t2 was defined as
Where s(t) represents the distanceof the object from some reference point.
The average velocity between two arbitrary instants of time, t and t+h, is given by the
difference quotient:
The velocity, v(t), is defined to be the limit of the average velocity as h approaches zero:

13. The velocity v(t) is rarely constant. The rate of change of the velocity is called
acceleration a(t) and is defined as
Example:
6. An object dropped from the roof of a 400-ft building. The object’s distance above
the ground, s(t), in feet, as a function of t, the time in seconds, is given by equation
s(t) = -16t2+400
The position of the objects for various values of t is shown in figure 2
a. Find the velocity as a function of t.
b. How fast is the object moving, when t=3 sec?

15. B. Time rate of change
If a variable t represents the time, the time rate of change of the function y=f(t)
with respect to t is described by the derivative of y with respect to t:
For exampe, the velocity is the time rateof change distance; acceleration is the time
rate of change of velocity.
Example:
7. annual profits of a newcomputer software company are given by the equation:
P(t)= -1 + 0.5t – 0.01t2
Where t represent the time (years)since the company began and P(T) represents
annual profits in millon of dollars. Find the rate which the company’s profit are
changing when t=5

17. C. Derivation of the Power Rule
If the function to be differentiated has the form f(x) = xn where n is a positive
integer, is derivative f ’(x) is

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25. Simplyfing yields
f'x=24x+32(3x2+8x-1)3
n
u(x)
n-1
u'(x)
n
u(x)
n-1
u'(x)

26. 
Last Operation
Differentiation
Rule
AdditionSubstraction
Sum Rule
Multiplication
Product rule
Division
Quotient rule
Raising to a power
Power rule

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