What is a Derivative?
The 5 Main Derivative Rules
Special Case Derivatives: Your new multiplication tables
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Example 1: Chain Rule Derivative

Differentiate f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4 with respect to x.

 

Step 1: Simplify and look for algebraic rewrites.

 

Here the equation is a simplified as we need it, and there are no algebra rewrites.

You always have to check though.

f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4

Step 2: Identify your primary rule as the chain rule.

 

When we look at this problem, we can see that it fits the general shape of a chain rule .

We have SOMETHING inside SOMETHING .

An equation inside another equation .

5 x 4 + 3 x 2 is inside (               ) 3 4 .

 

f ( x ) = ( SOMETHING ) Something

f ( x ) = ( IN ) OUT

f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4

Step 3: Break the problem up into two these two bite-size problems.

IN = SOMETHING and OUT = SOMETHING

 

IN = 5 x 4 + 3 x 2                 OUT = ( 5 x 4 + 3 x 2 ) 3 4

Step 4: Take the derivatives of those two bite-size problems to find your DIN and your DOUT .

 

The derivatives of IN and OUT only require us to apply the power rule.

Bring the power down.

Subtract 1 from the power.

 

DIN : The derivative of the IN behaves like a normal power rule derivative that you have done many times before.

 

DOUT : When you are taking the derivative of the OUT piece, you ignore what is inside the parenthesis., and perform the derivative rule only for the outside piece, and leave the IN alone.

IN = 5 x 4 + 3 x 2                 OUT = ( 5 x 4 + 3 x 2 ) 3 4

DIN = 4 5 x 4 1 + 2 3 x 2 1                 DOUT = 3 4 ( 5 x 4 + 3 x 2 ) 3 4 1

DIN = 20 x 3 + 6 x 1               DOUT = 3 4 ( 5 x 4 + 3 x 2 ) 1 4

DIN = 20 x 3 + 6 x 1               DOUT = 3 4 ( 5 x 4 + 3 x 2 ) 1 4

 

 

 

Step 5: Bring it all back together following the chain rule recipe: (DIN)(DOUT).

 

f ( x ) =   ( DIN ) ( DOUT )  

f ( x ) =   ( 20 x 3 + 6 x ) 3 4 ( 5 x 4 + 3 x 2 ) 1 4

 

Final Result:

The derivative of f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4  is f ( x ) =   ( 20 x 3 + 6 x ) 3 4 ( 5 x 4 + 3 x 2 ) 1 4 .

Most of the time you will want to leave the derivative as it is and not try to expand it or simplify it.

My rule of thumb is if someone wants the derivative give them this unless they ask you to simplify further.

If you need to do more work, like finding a second derivative, you would want to consider expanding and simplifying to make your life easier.

 

Meaning:

          The equation for finding the slope of any tangent line at any x-value of f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4   is f ( x ) =   ( 20 x 3 + 6 x ) 3 4 ( 5 x 4 + 3 x 2 ) 1 4 .

 

          The instantaneous rate of change for every x-value of f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4 , is found by using the derivative equation, f ( x ) =   ( 20 x 3 + 6 x ) 3 4 ( 5 x 4 + 3 x 2 ) 1 4 .

 

          The slope of f ( x ) = ( 5 x 4 + 3 x 2 ) 3 4 at any single x-value can be found by plugging it into the derivative, f ( x ) =   ( 20 x 3 + 6 x ) 3 4 ( 5 x 4 + 3 x 2 ) 1 4 .

 

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