Application of the Product Rule

In summary, Daniel is explaining that the product rule can be used to solve problems in which two terms are multiplied together. The first step is to identify the two terms in the product and apply Leibniz' rule. The second step is to factor the equation. Once the equations have been factored, the answer can be found.
  • #1
Samael
11
0
A problem which I'm sure is rather simple, however I cannot seem to simplify the equation properly to produce the fully simplified answer as written in my textbook.

The problem being.

[tex]f(x)= e^{4x}{(1-2x)}^4[/tex]

Find f ' (X)


All help is greately appreciated!
 
Last edited:
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  • #2
Are you supposed to take the derivative?
From the title, you already know you can use the product rule. Show us what answer you got.
 
  • #3
Yes, find the derivative using the PR. The problem I'm having is that I cannot seem to simplify the problem after applying the rule.

The answer to the above problem was:

[tex]-4e^{4x}{(1-2x)}^3(2x+1)[/tex]

However a fully worked solution would explain what went on.
 
  • #4
You can't simplify more than identifying the 2 terms in the product and apply Leibniz' rule...

BTW,you'll need some algebra afterwards to get to their answer.

Welcome to Physicsforums and,on behalf of the crew,I thank you for the trust and the
support for this forum.:smile:

Daniel.
 
  • #5
That algebra means factoring the common pieces in the sum you get.

HINT:the exponential and the polynomial with the degree "3".

Daniel.
 
  • #6
dextercioby said:
You can't simplify more than identifying the 2 terms in the product and apply Leibniz' rule...



Daniel.

I don't think I've ever heard, or been taught that rule before. Would you be able to elaborate on it?
 
  • #7
It's the product rule,invented by Gottfried Wilhelm Leibniz around 1780.

[tex] (ab)'=a'b+ab' [/tex]

Daniel.
 
  • #8
Ok. However occording to the rule the problem should become:

[tex] e^{4x} .-8{(1-2x)}^3 + {(1-2x)}^4. 4e^{4x} [/tex]

Where to go from here, I am not so sure.
 
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  • #10
Sorry, I placed a totally different problem and had to go back and fix it.
 
  • #11
Here's what I'm getting

[tex] f'(x)=\left(e^{4x}\right)'(1-2x)^{4}+e^{4x}\left[(1-2x)^{4}\right]'
=4e^{4x}(1-2x)^{4}-8e^{4x}(1-2x)^{3} [/tex]

Now do what i said,factor the common parts.

Daniel.
 
  • #12
I'm not familar with that form of the product rule. I was thinking of:
[tex] uv'+vu'[/tex]
 
  • #13
It's not smart to switch between the variables (change their order) in the product.We physicists never do it.

Daniel.
 
  • #14
That is the Rule we have been taught, although your representation of it now makes a whole lot more sense than the method we are told to use. Thanks for that.

Also factorising your equation gave:
[tex](4e^{4x} - 8e^{4X} (1-2x)^3

=

-4e^{4x}(1-2x)^3
[/tex]
 
Last edited:
  • #15
Nope.

[tex] 4e^{4x}(1-2x)^{3}\left[(1-2x)-2\right]=-4e^{x}(1-2x)^{3}(1+2x) [/tex]

Daniel.
 
  • #16
Samael said:
That is the Rule we have been taught, although your representation of it now makes a whole lot more sense than the method we are told to use. Thanks for that.

Also factorising your equation gave:
[tex](4e^{4x} - 8e^{4X} (1-2x)^3

=

-4e^{4x}(1-2x)^3
[/tex]

Your representations of the product rule are the same.
 
  • #17
Thanks for the assistance, its much appreciated. :)
 

Related to Application of the Product Rule

What is the Product Rule?

The Product Rule is a formula used in calculus to find the derivative of a function that is the product of two other functions. It states that the derivative of a product is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

How is the Product Rule used in real-world applications?

The Product Rule is used in many real-world applications, such as calculating the rate of change in physics and engineering problems, determining the growth rate of populations in biology, and finding the optimal production levels in economics. It is also used in finance to calculate compound interest and in statistics to find the expected value of a product of two random variables.

What are the steps for applying the Product Rule?

The steps for applying the Product Rule are as follows:

  1. Differentiate the first function, keeping the second function the same.
  2. Then, differentiate the second function, keeping the first function the same.
  3. Multiply the first function by the derivative of the second function.
  4. Add this to the product of the second function and the derivative of the first function.
  5. Simplify the resulting expression, if possible.

Can the Product Rule be used for more than two functions?

Yes, the Product Rule can be extended to more than two functions. For example, if you have a function that is the product of three functions, you would differentiate the first function, keeping the other two functions the same, then differentiate the second function, keeping the first and third functions the same, and finally differentiate the third function, keeping the first two functions the same. You would then multiply each of these derivatives by the other two functions and add them together to get the final result.

When should the Product Rule be used instead of the Quotient Rule?

The Product Rule should be used when you are differentiating a function that is the product of two other functions. The Quotient Rule, on the other hand, should be used when you are differentiating a function that is the quotient of two other functions. It is important to choose the correct rule based on the given function in order to get an accurate result.

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