Simplifying the Product Rule for Derivatives

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Discussion Overview

The discussion focuses on finding the derivative of the function g(x) = (4x² - 2x + 1)e^x, specifically addressing the application of the product rule and the simplification of the resulting expression.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents their derivative calculation using the product rule but is uncertain about its correctness.
  • Another participant suggests that while the initial calculation is not wrong, it can be simplified by factoring.
  • Subsequent replies express confusion about how to simplify the expression through factoring.
  • A later reply indicates that there is a common factor in the result that can be identified and factored out.
  • One participant attempts to simplify the expression and presents a new form, but questions whether it is indeed simplified.
  • Another participant provides a corrected and fully factored version of the derivative, prompting a clarification about a previous typo.
  • The discussion concludes with a light-hearted acknowledgment of the simplification achieved.

Areas of Agreement / Disagreement

Participants generally agree that the initial derivative calculation is not incorrect, but there is no consensus on the best method for simplification, as some express uncertainty about the factoring process.

Contextual Notes

There are unresolved aspects regarding the clarity of the factoring process and the participants' understanding of simplification techniques.

lastochka
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Hello,

I have this exercise that I can't get the right answer. I have to find derivative of

g(x)= (4${x}^{2}$-2x+1)${e}^{x}$

So, what is did is

g$^{\prime}$=(8x-2)${e}^{x}$+(4${x}^{2}$-2x+1)${e}^{x}$

My Prof said it is wrong... I am not sure if I have to multiply the brackets or what I did is completely wrong.
Can someone, please, check it for me?
Thank you!
 
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Technically, it is not wrong, however, it can be simplified by factoring...what do you get when you fully factor?
 
Thank you for answering! I am not sure how to simplify it by factoring...
 
lastochka said:
Thank you for answering! I am not sure how to simplify it by factoring...

There is a factor common to both terms in your result...can you spot it?
 
Yes, I see that, but will it make it simplified...
Here is what I have
${e}^{x}$(4${x}^{2}$-6x-1)
Is that it?
Thank you for helping!
 
You are close...here's what I get:

$$g'(x)=(8x-2)e^x+\left(4x^2-2x+1\right)e^x=e^x\left(8x-2+4x^2-2x+1\right)=e^x\left(4x^2+6x-1\right)$$
 
Oh, sorry I mistype, it is plus 6x not minus...
Thank you so much for helping!
 
Good deal! Wouldn't you say that is simpler than the unfactored version? :D
 

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