Derivative Notation

The notation for derivative just like a function:

f’(x)

For derivative, there are slashy dot thing is called a prime. The expresion of it is f prime of x. The other notation is y’ (y prime), because f(x) and y are interchangeable. It also same meaning with

, so the notations of derivative are:

f’(x) = y’ =

is meaning

It is the key to understanding derivative:

Derivative is slope of tangent line at point. Then remember that the slope of a line is

For the derivative we need 2 point:

1. (x,f(x))

(x,f(x)) is same meaning with (x,y), because y replace by f(x).

2. ((h+x),f(h+x))

h is change in x.

we can slope of line between 2 points into the slope formula

the slope formula is

Subtitute the two points (x,f(x)) ,((h+x),f(h+x)) in to the slope formula:

x is cancelled out by –x, so we find that

The definition of derivative is

f’(x)=

we can read it “f prime x is the limit f(x plus h) minus f(x) all divide by h, h approach zero.

So, f’(x) is the limit of slope, but if we look closer, we will know that it’s also limit change in values of f(x) over change in x. It’s also instantaneous rate of change.

Let's do this exercise:

Given: f(x)= 4x²-8x+3

Find: f’(2)

Let’s follow the prosedure to solve this question:

To slope this question, first thing we do is subtitute our x in the derivative definition. In our problem f’(2) is change x by 2. So, subtitute 2 for x in the derivative definition.

f’(x)=

, x=2

f’(2)=

To get f(2), subtitute 2 into the given function

f(2)= 4(2)²-8(2)+3

= 3

So we find x and y coordinates is (2,3), because f(2)=3

To complete the formula, we need f(2+h). Then subtitute (2+h) in to the given function

f(2+h)= 4(2+h)²-8(2+h)+3

= 4(4+4h+4h²)-8(2+h)+3

= (16+16h+16h²)-(16 + 8h)+3

= 4 h²+8h+3

To get derivative, subtitute values of f(2) and f(2+h) in to derivative formula

f’(2)=

Remember, we can’t subtitute zero for h because we get zero in denominator. And remember, if we doing limits, if you can’t subtitute , try simplifying or factoring.