Differential of Exponential Function [Verify My Answer]

Jacob

Jacob

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f(x) = 4x^3 e^x

U = 4x^3 , V = e^x
U' = 12x^2 , V' = e^x

(4x^3) (e^x) + (e^x) (12x^2)
(4x^3 e^x) + ( 12x^2 e^x)
Thus, 4e^x ( x^3 + 3x^2)<-- is this answer correct?
I know 4(x^3e^x + 3x^2e^x) is the correct answer. But the one I get above looks the same to me or I really miss out something?

Also for another question.

f(x) = 5lnx / x^3

U = 5lnx , V = x^3
U' = 5/x . V' = 3x^2

(x^3)(5/x) - (5lnx) (3x^2) / (x^3)^2
5x^2 - 3x^2 (5lnx) / x^6
Thus, 5 - 15 ln x / x^4
 
Last edited:
Jacob said:
f(x) = 4x^3 e^x

U = 4x^3 , V = e^x
U' = 12x^2 , V' = e^x

(4x^3) (e^x) + (e^x) (12x^2)
(4x^3 e^x) + ( 12x^2 e^x)
Thus, 4e^x ( x^3 + 3x^2)<-- is this answer correct?
I know 4(x^3e^x + 3x^2e^x) is the correct answer. But the one I get above looks the same to me or I really miss out something?

Also for another question.

f(x) = 5lnx / x^3

U = 5lnx , V = x^3
U' = 5/x . V' = 3x^2

(x^3)(5/x) - (5lnx) (3x^2) / (x^3)^2
5x^2 - 3x^2 (5lnx) / x^6
Thus, 5 - 15 ln x / x^4
Click to expand...
f(x) = 4x^3 e^x

f'(x) = 4e^x ( x^3 + 3x^2)<-- is this answer correct? .......Yes .... I would have further factorized to:

f'(x) = 4 * e^x * x^2 * ( x + 3)

2) f(x) = 5lnx / x^3

f'(x) = 5 - 15 ln x / x^4 ................... as written it is incorrect.

It should be written as (with grouping symbols):

f'(x) = [5 - 15 * ln(x)] / x^4

Otherwise - good work.
 
Your first answer is correct; theirs is just written in a different form.

Oddly, both are only partly factored. If you want it unfactored, then your initial form, 4x^3 e^x + 12x^2 e^x, makes most sense; if you want it factored, you can factor out even more than you did, and write 4x^2 (xe^x + 3e^x).

None of these is wrong!

Similarly, your second answer, when written with required parentheses as (5 - 15 ln x) / x^4, is correct; if you chose, you could factor out a 5 and write it as 5(1 - 3ln x)/x^4.
 
I have no idea why I said what I did!

Well, actually, I do. I started by pasting in the unfactored form, but was thinking as if it was the partially factored form. Then I forgot to read what I wrote. In other words, always check your answer ...
 
Thank you guys so much for helping me checking my answers. yeah, I actually had brackets in my notes but I forgot to add them when I typing in.
 
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