Differentiation In Mathematics

The Mathematics lesson derivative topic. Calculating and definition of the derivative. Relationship of the derivative and limit. Finding the derivative using limit. The subject expression and solved examples.


In a function limit value of ratio of a changing which occurs in a dependent variable to a changing which occurs in a independent variable is called the derivative.

Let we consider a function that in f(x) = y form.


Because of to be y = f(x), the changing that occurred in variable y by the increase of the independent variable is found as the following.

Δy = f(x + Δx) – f(x)

If we does change of Δx in the variable x in the function f(x), the change occur in the y is shown at the above. This changing is shown in the Δy form.

Due of to be Δy = f(x + Δx) – f(x)

Ratio of Δy
Δx 




is shown in the form f(x + Δx) – f(x)
Δx 




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The limit above is called the derivative of the function f(x) .

In the above expression the value of the limit must be a real number. Otherwise there is no derivative.

If we accepted Δx = h, the derivative is given in the following form.

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Definition of The Derivative

A⊂ R, a ϵ A

Function f is a function defined in cluster A

The limit of the 

Limx → a    [f(x) – f(a)]
x – a




or putting a + h instead of x

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If there is the limit, the function f has a derivative at the point a.

If function f has the derivative at point a, the function f is a function that can be derived at the point a.

The derive of a function can be shown various form. The forms in the following is the derivative of the function f.

f’(x),df(x) 
, Df(x),dy
dx
dx




Right-Sided Derivatives

If the variable x draw near only from values the bigger than z to the value z, in this case derivative of the function f are called the right-sided derivative.

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In the limit above the variable x is nearing the point a from the values bigger than a.

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In the limit above the h variable is nearing the value 0 from the values bigger than 0.


The Left-Sided Derivatives

If the variable x is nearing the point a from the values smaller than the a or the variable h is nearing the point 0 only from values smaller than 0, these derivates are called the left-sided derivatives.

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If a function has the left-sided and right-sided derivatives and these derivatives are equal to each other, this function is a can be derived function.


Examples of the Derivatives

Now let we do the derivative calculations using the limit expression.


Example – 1 

Function f(x) has given below

f(x) = 6x2 + 5x – 1 

Find the derivative (f’(x) ) of this function.


Solution:

We're asked to find the value of 

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We will do the operations of the multiplication and simplification out of limit and we will get the simplest form of these operations into the limit.

f(x+h) = 6. (x+h)2 + 5.(x+h) -1

f(x+h) = 6(x2 + 2xh + h2) + 5x + 5h – 1

f(x+h) = 6x2 + 12xh + 6h2 + 5x + 5h – 1

f(x+h) – f(x) = 6x2 + 12xh + 6h2 + 5x + 5h – 1 – 6x2 – 5x + 1

f(x+h) – f(x) = 6h2 + 12xh + 5h

f(x+h) – f(x) 
= 6h2 + 12xh + 5h
h
h




f(x+h) – f(x) = 6h + 12x + 5
h





Let we get this result into the limit.

Limh→0 [f(x + h) – f(x)
h




= Limh→0 ( 6h + 12x + 5)

= 6.0 + 12x + 5 

= 12x + 5


Example – 2 

Function f(x) has given below

f(x) = 3x2 – 5x + 12 

Find the result of operation d(f)/d(x) using limit.


Solution:

f’(x) = Limh→0 f(x + h) – f(x)
h




First let we find the expression of f(x + h). Next, let we calculate the expression of f(x + h) – f(x). Next, we will do the operation of f(x + h)/f(x). Finally, we will get the limit of the value of f(x+h)-f(x)/h.


f(x) = 3x2 – 5x + 12

To find the expression of the f(x+h), it is written (x + h) instead of all x.

f(x + h) = 3(x + h)2 – 5(x + h) + 12

f(x + h) = 3(x2 + 2xh + h2) – 5x – 5h + 12

f(x + h) = 3x2 + 6xh + 3h2 - 5x – 5h + 12

Now, We will find the expression of f(x + h) – f(x).

f(x + h) – f(x) = 3x2 + 6xh + 3h2 - 5x – 5h + 12 – 3x2 + 5x – 12

f(x + h) – f(x) = 3h2 + 6xh – 5h


Let we do the operation of f(x + h) – f(x) /h

f(x + h) – f(x)= 3h + 6x – 5
h




Now, let we get this result into the limit.

f’(x) = Limh→0 f(x + h) – f(x)
h



= Limh→0 (3h + 6x – 5)

= 3.0 + 6x – 5  

= 6x – 5


Example – 3 

Calculate the derivative of function f(x) = 2x3 + 8x –2 at the point of x = 0 using limit.


Solution:

We will use definition of the limit for this question.

The derivative at the point a of a function can be find as the following.

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For a = 0,

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f(x) – f(0) =2x3 + 8x –2 – (0 + 0 – 2)

f(x) – f(0) = 2x3 + 8x


f(x) – f(0) 
=2x3 + 8x
x
x





f(x) – f(0)  = 2x2 + 8
x




Let we put this result into the limit.

= Limx→0 (2x2 + 8)

= 0 + 8

= 8

f’(0) = 8


Division Rules In Derivates




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January 18 2019

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