PREFACE
Thank to the Almighty God which has given a healthy favor and had the grace and blessings Us, so We could finish the paper this had the title “DERIVATIVE ”. And We would like to thank all Our friends who have been taking part in the making of this paper and also to all those who helped in the completion of this paper.
For the perfection of this paper, we expect some criticism and suggestions from readers as We only human who make mistakes. Hopefully, This paper could help the readers to expand their knowledge about Calculus especially about Derivative.
Tondano, 14th February 2015
8th Group
Preface……...1 Table Of Contents ...2 CHAPTER I Introduction
A. Background...3 B. Purpose…...3 CHAPTER II Discussion
A. The
Derivative...4 B. Notation of Derivative...10 CHAPTER III Closing
A. Conclusion...11 B. References...12
A.BACKGROUND
In this paper our group will explain about Derivative which is part of calculus. The concept of Derivative is at the core of Calculus and modern mathematics. The definition of the derivative can be shown in two different ways. One is geometrical (as a slope of a curve) and the other one is physical (as a rate of change).
B.PURPOSE
1. To know about “The Definition Of Derivatives”
2. Find out Determining Derivative of a Function
3. To use the definition of derivative to find the derivative of several functions 4. To understand the relationship between differentiability and continuity.
The Defininoi of ihe Deinatinae
In the last discussion of 7th Group about “Two Problems - One Theme” , we saw that the computation of the slope of a tangent line, and the instantaneous velocity of an object at x=a all required us to compute the following limit.
f'(x)=lim
x → a
f(x)−f(a) x−a
The form of the function above is
Equivalent Form
of the Derivative. In using of letter that used above, there is no Absolute Rule that manage DerivativeNotation.
This kind of function can be converted into another form, such as,
f'(x)=lim
h →0
f(a+h)−f (a) h
We will use this form along this discussion. This is such an important limit and it arises in so many places . We call it a derivative. Derivative can be define in official definition like below.
Example 1 Find the derivative of the following function using the definition of the
derivative.
f(x)=2x2−16x+35
Solution
To determine the Derivative of the function above, the first and important thing is substitutes it into the definition we have got before.
f'(x)=lim
Example 2 Find the derivative of the following function using the definition of the
derivative.
g(t)=t+t1
Solution
In this example, some step of algebra was ‘Jumped’. We do that thing to make our work easier and efficient, but It would not be harder than previous example.
And just like before, First, We plug the function into the definition of the
things a little. In this case we will need to combine the two terms in the numerator into a single rational expression as follows.
g' the denominator. Multiplying out the denominator will just overly complicate things so let’s keep it simple. Next, as with the first example, after the simplification we only have terms with h’s in them, left in the numerator and so we can now cancel an h out.
Example 3 Find the derivative of the following function using the definition of the
derivative.
R(z)=√5z−8
Solution
R'(z)=lim At Infinity”. We CANNOT just substitutes h=0 into the function. So, the best solution
is, We have to determine and compare at The Two-One Sided Limits.
Beside it, the absolute value will help us to determine The Two-One Sided Limits by its definition that written as:
because
h>0in a right-hand limit.
The two one-sided limits are different , So We can conclude
lim h→0
|h|
hdoesn’t exist.
if ‘THE LIMIT DOES NOT EXIST’ then ‘THE DERIVATIVE DOES NOT EXIST EITHER’
In this example we have finally seen a function for which the derivative doesn’t exist at a point. Derivatives will not always exist. In fact, the derivative of the absolute value function exists at every point EXCEPT the one we just looked at x=0
Definition
A function f(x) is called DIFFERENTABLE at x=a, if f '(a) exists
And
f(x) is called DIFFERENTABLE on an interval, if the derivative exists for each point
in that interval.
And next, We will see a Theorem that show Us a relationship between continuous functions with a differentiable functions.
Theorem
If f(x) is differentiable at x=a then f(x) is continuous at x=a.
Note that this theorem DOES NOT work in reverse. Consider f(x)=|x| and take a look at lim
x →0f(x)=limx →0|x|=0=f(0)
So, f(x)=|x| is continuous at x=0 but we’ve just shown above in Example 4
NOTATION OF DERIVATIVE
Next, We need to discuss some alternate notation for the derivative. The typical derivative notation is the “prime” notation. However, there is another notation that is used on occasion.
Given a function y=f(x) all of the following are equivalent and represent the
derivative of f(x) with respect to x.
f'
(x)=y'=dfdx=dydx=dxd (f(x))=dxd (y)
Note as well that on occasion we will drop the (x) part on the function to simplify the notation somewhat. In these cases the following are equivalent.
f'
(x)=f '
As a final note in this section, We will conclude that computing most derivatives directly from the definition is a fairly complex (and sometimes painful) process filled with opportunities to make mistakes. In a couple of sections We will start developing formulas and/or properties that will help us to take the derivative of many of the common functions so we won’t need to resort to the definition of the derivative too often.
CHAPTER III CLOSING
CONCLUSION
From the discussion of the derivative, we can concluded: 1. Definition of derivative
The derivative of f (x) with respect to x is the function f’(x) and is defined as,
f '(x)=lim
h →0
f(x+h)−f (x) h
2. Definition of
DIFFERENTABLE
function.A function f(x) is called DIFFERENTABLE at x=a, if f '(a) exists
And
f(x) is called DIFFERENTABLE on an interval, if the derivative exists for each point in
that interval.
3. Continuity Of Derivative Function
If f(x) is differentiable at x=a then f(x) is continuous at x=a.
4. Notation of derivative
f'
(x)=y'=dfdx=dydx=dxd (f(x))=dxd (y)
