basic calculus help

[Pantlegz]

it's pretty basic but I don't really know where to start with it.

y'=((e^αe)^-2&#945😉

I know derivatives with exponents would be something like

-2((e^αe)^-2&#945😉)(-2&#945😉'
-2(α((e^αe)^2&#945😉)(αe)')(-2)
-2(α((e^αe)^2&#945😉)(1))(-2)

But it doesn't look right, I'm sure I really messed something up with chain rule, we're also working with product rule, which also isn't applied. So I'm sure that's nowhere near correct.

 

[Wizlem]

What is the question? Are you trying to find out what y is? Is y a function of a? Also is that supposed to be written that way or do you mean y'=e^(ae^(-2a))? The version you have seems to be equal to e^(-2(a^2)e)

 

[Wizlem]

I see your trying to do the derivative so maybe thats what you are asking. If its written correctly now and y is a function of a, the answer is (-4ea)((e^ae)^-2a).

 

[Pantlegz]

yea I'm trying to find the detivative for the problem, the way it's written in the book is:

y=e^αe^-2α. which isn't turning on the way I want it to so I'll type it out, it's e to the ae power to the -2a power. If that makes sense.

So I simplified it the way my professor told me to which is what I have above. If that's there you got (-4ea)((e^ae)^-2a), would you mind breaking it down so I can see where I'm off. Thanks!

 

[yottabit]

Here's my take:

 

[edro]

Why on earth do they teach calculus without story problems?
No one can relate to a bunch of equations...

 

[Belegost]

So, like Wizlem, I'm slightly confused by your notation, since y(a)=((e^αe)^-2&#945😉 simplifies to e^(-2a^2e) but I honestly feel that the desired function is y(a) = e^(ae^(-2a)).

So I'll solve for that problem.
Let f(a) = e^(-2a) then, f'(a) = -2e^(-2a)
y(a) = e^(af(a)) => y'(a) = e^(af(a))*d/da(af(a))
and d/da(a*f(a)) = f(a) + af'(a)

Combining:
y'(a) = e^(ae^(-2a))[e^(-2a) -2ae^(-2a)]
Simplifying:
y'(a) = e^(a(e^(-2a) - 2)[1-2a]

To give our friend edro a story:

Billy has a continuous population that grows (approximately) exponentially in time (denoted a) The growth constant itself decays exponentially with time with constant -2. Billy really wants to know the instantaneous rate of growth at any given time, give him the necessary function of a.