What is a Derivative?
The 5 Main Derivative Rules
Special Case Derivatives: Your new multiplication tables
1 of 2

Example 1: Limit Definition of the Derivative

Find the derivative,   f ( x ) , of

f ( x ) = 5 x 2 + 3 x + 1

 using the limit definition of the derivative.

Step 1: Know the Limit Definition of the Derivative.

 

f ( x ) = lim h 0 f ( x + h ) f ( x ) h

Step 2: Find the  f ( x ) piece.

This was our given equation.

f ( x ) = lim h 0 f ( x + h ) f ( x ) h

 

f ( x ) = 5 x 2 + 3 x + 1

 

Step 3: Find the f ( x + h ) piece.

Apply the function notation process which is telling you to plugin x + h everywhere you see an x in the original equation, f ( x ) . You will need to expand this portion of the equation.

 

Function notation works the same no matter if we are given an actual number to plugin to the equation, or, like we are here, given an equation to plug in.

 

Whatever they replace the x with inside the parenthesis of f ( x ) , you replace all the x ’s within the equation. Normally you see this when some asks you to find f ( 5 ) , which is asking you to plug 5 in wherever you see an x in the equation.

 

The process is the same with f ( x + h ) , you are being told to plug in ( x + h ) everywhere you see an x in f ( x ) .

 

From there you will expand f ( x + h ) . This will normally mean expanding out any squared or cubed terms using distribution (you might call it FOIL).

f ( x ) = lim h 0 f ( x + h ) f ( x ) h

 

f ( x ) = 5 x 2 + 3 x + 1

f ( x + h ) = 5 ( x + h ) 2 + 3 ( x + h ) + 1

f ( x + h ) = 5 ( x + h ) 2 + 3 ( x + h ) + 1

f ( x + h ) = 5 ( x + h ) ( x + h ) + 3 x + 3 h + 1

f ( x + h ) = 5 ( x 2 + 2 xh + h 2 ) + 3 x + 3 h + 1

f ( x + h ) = 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1

 

f ( x + h ) = 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1

Step 4: Plug the  f ( x ) piece and the f ( x + h ) piece into the limit definition of the derivative.

 

A critical part of this step is to always put the f ( x ) piece in a set of parentheses behind minus sign. You are subtracting the entire f ( x ) piece.

 

f ( x ) = lim h 0 f ( x + h ) f ( x ) h

f ( x ) = lim h 0 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1 ( 5 x 2 + 3 x + 1 ) h

 

Step 5: Simplify the equation as much as possible, and then treat it like any other limit problem and run the limit process.

 

In this example (and in a lot of problems) the first move is to distribute the negative sign through the parenthesis in the back of the problem. From there you should see some great cancelation.

 

Not always, but a lot of the time, you will see that you can factor an h out of what is left on the top of the fraction. From there you are able to cancel the h on the bottom of the fraction, and you are ready to try Option 1: Plug It In.

 

No matter the limit process you must apply, the final result will usually be the canceling of the h on the bottom of the limit definition equation.

f ( x ) = lim h 0 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1 ( 5 x 2 + 3 x + 1 ) h

f ( x ) = lim h 0 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1 5 x 2 3 x 1 h

f ( x ) = lim h 0 5 x 2 + 10 xh + 5 h 2 + 3 x + 3 h + 1 5 x 2 3 x 1 h

f ( x ) = lim h 0 10 x h + 5 h 2 + 3 h h

f ( x ) = lim h 0 ( 10 x + 5 h + 3 ) h

f ( x ) = lim h 0 h   ( 10 x + 5 h + 3 ) h  

Option 1: Plug It In f ( x ) = lim h 0 10 x + 5 h + 3 = 10 x + 5 ( 0 ) + 3

f ( x ) = 10 x + 3

 

 

 

 

Final Result:

The derivative of f ( x ) = 5 x 2 + 3 x + 1  is  f ( x ) = 10 x + 3

 

Meaning: The equation for finding the slope of any tangent line of f ( x ) = 5 x 2 + 3 x + 1   is f ( x ) = 10 x + 3 .

 

Post a comment

Leave a Comment

Free to Use!

X