Simplifying the Product Rule for Derivatives

In summary, the conversation discusses finding the derivative of g(x) = (4x^2 - 2x + 1)e^x and simplifying it by factoring. The final answer is g'(x) = e^x(4x^2 + 6x - 1).
  • #1
lastochka
29
0
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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  • #2
Technically, it is not wrong, however, it can be simplified by factoring...what do you get when you fully factor?
 
  • #3
Thank you for answering! I am not sure how to simplify it by factoring...
 
  • #4
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?
 
  • #5
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!
 
  • #6
You are close...here's what I get:

\(\displaystyle g'(x)=(8x-2)e^x+\left(4x^2-2x+1\right)e^x=e^x\left(8x-2+4x^2-2x+1\right)=e^x\left(4x^2+6x-1\right)\)
 
  • #7
Oh, sorry I mistype, it is plus 6x not minus...
Thank you so much for helping!
 
  • #8
Good deal! Wouldn't you say that is simpler than the unfactored version? :D
 

FAQ: Simplifying the Product Rule for Derivatives

What is the Product Rule?

The Product Rule is a mathematical rule used to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

Why is the Product Rule important?

The Product Rule is important because it allows us to find the derivative of more complex functions that are made up of products of simpler functions. It is a fundamental rule in calculus and is used in many real-world applications, such as physics, engineering, and economics.

How do you use the Product Rule?

To use the Product Rule, you first identify the two functions that are being multiplied together. Then, you take the derivative of each function separately. Finally, you plug these derivatives into the Product Rule formula to find the derivative of the entire product function.

Are there any shortcuts for using the Product Rule?

Yes, there are a few shortcuts for using the Product Rule. One is the "Leibniz notation" which allows you to easily remember the formula for the Product Rule as (u * v)' = u' * v + u * v'. Another shortcut is using the "prime notation" where the derivative of a function is denoted by putting a prime symbol (') after the function.

Can the Product Rule be applied to more than two functions?

Yes, the Product Rule can be extended to any number of functions. In this case, the formula becomes (f1 * f2 * f3 * ...)' = f1' * f2 * f3 * ... + f1 * f2' * f3 * ... + f1 * f2 * f3' * ... + ...

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