Anybody good with calculus product rule?

(x-2)e^(2-x)

Question

Anybody good with calculus product rule?

(x-2)e^(2-x)

mertsj · Answer

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

mertsj · Answer

Assuming that you want the first derivative.

anonymous · Answer

Where does the (-1) come from??

.sam. · Answer

comes from chain rule.

turingtest · Answer

what is the derivative of$$2-x$$?

anonymous · Answer

But I thought e differentiates to itself?

turingtest · Answer

but you need to use the chain rule as .Sam. said....

turingtest · Answer

$${d\over dx}e^{2-x}=e^{2-x}\cdot{d\over dx}(2-x)=-e^{2-x}$$

turingtest · Answer

@shallster are you familiar with the chain rule?
or is the post above confusing you?

anonymous · Answer

pretty confusing.

turingtest · Answer

ok, can you take the derivative of$$(2-x)^2$$?

anonymous · Answer

it's not asking me to do that though??
The equation is (x-2)e^(2-x)
I need the derivatives of x-2 & e^(2-x) don't I??

turingtest · Answer

yes, but you seem to have trouble differentiating$$e^{2-x}$$so I'm trying to help you do so

anonymous · Answer

but i thought e differentiates to itself?

turingtest · Answer

again: you $must$ use the chain rule here
$$\frac d{dx}e^x=e^x$$yes, but this is not the case here, we have a compound function so we must use the chain rule

anonymous · Answer

You obviously know what you're talking about, so I'm all ears. I NEED help.

turingtest · Answer

so again I ask you to demonstrate your understanding of the chain rule
what if I asked you to differentiate$$(1-x^2)^2$$by using the chain rule, could you do that?

anonymous · Answer

4x(1-x^2) ???

turingtest · Answer

exactly, you had to differentiate the inside function as well
now that I know you understand this I can explain the other derivative...

turingtest · Answer

$$\frac d{dx}e^{f(x)}=e^{f(x)}\cdot f'(x)$$see?
you still leave the e^(whatever) part the same, but you still must differentiate the second function, as always in the chain rule

turingtest · Answer

do you agree with the above line?
let me know what is confusing you if not

anonymous · Answer

so e^(2-x) = -1e^(2-x)

turingtest · Answer

yep :)
now does mertsj's answer make sense?

anonymous · Answer

I've got it down to e^2-x) - (x-2)e^(2-x) and I'm not that sure how to simplify it.

turingtest · Answer

dang lag...
factor out $e^{2-x}$ like so$$(x-2)e^{2-x}$$$$e^{2-x} -(x-2)e^{2-x}=e^{2-x}(1-x+2)=e^{2-x}(3-x)$$so I think mertsj was a little bit off

anonymous · Answer

I also struggle with factoring. I'm such a douche.

turingtest · Answer

no, this kind of factoring is a bit tricky
I learned it through problems like these
as you can see, mertsj made a small mistake in factoring as well, despite his illustrious abilities
it happens to the best of us ;)

PS: as a mod, I have to ask you not to say the "d" word above
I'd like to keep you around after all!

anonymous · Answer

Understood.

mertsj · Answer

I can't believe I did that.  But, shallster, as you will learn, Turing is the resident expert.  So always pay attention to him.

anonymous · Answer

That I will.

turingtest · Answer

thanks y'all !
you guys make the site what it is though
agape love =)

mertsj · Answer

Right back to you, Turing.

anonymous · Answer

Still not sure how the -2 became +2 when factoring?

mertsj · Answer

By distributing the -1

mertsj · Answer

$$e ^{2-x}-(x-2)e ^{2-x}=e ^{2-x}[1-(x-2)]=e ^{2-x}[1-x+2]=e ^{2-x}[3-x]$$

anonymous · Answer

I understand the rest but how does (1-(x-2)) expand to (1-x+2)?

turingtest · Answer

distribute -1

anonymous · Answer

I bet it's something really simple that my brain is bypassing!!

turingtest · Answer

like that^ ?
:)

turingtest · Answer

$$-(x-2)=-x+2$$no?

anonymous · Answer

ah, i see. Thanks guys.