Simplifying the Product Rule for Derivatives

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SUMMARY

The discussion focuses on simplifying the derivative of the function g(x) = (4x² - 2x + 1)e^x. The initial derivative calculated was g'(x) = (8x - 2)e^x + (4x² - 2x + 1)e^x, which was deemed incorrect by the professor. The correct simplification involves factoring out e^x, resulting in g'(x) = e^x(4x² + 6x - 1), which is a more concise expression.

PREREQUISITES
  • Understanding of the Product Rule in calculus
  • Familiarity with exponential functions, specifically e^x
  • Basic algebraic factoring techniques
  • Knowledge of derivatives and their notation
NEXT STEPS
  • Study the Product Rule for derivatives in detail
  • Practice factoring polynomials to simplify expressions
  • Explore the properties of exponential functions and their derivatives
  • Learn about common mistakes in derivative calculations and how to avoid them
USEFUL FOR

Students studying calculus, particularly those learning about derivatives and simplification techniques, as well as educators looking for examples of common errors in derivative calculations.

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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