Any calc gurus around tonight? I’m proofreading a calculus textbook, and though I’m not responsible for the math, I am querying things that don’t look right (as far as I can tell from my 20-years-ago calc classes). Here’s the latest, and I’d just like a backup (please excuse the notation, I don’t have the time to figure out how to make it pretty; bold is my emphasis):
We are simplifying:
([-1/2]t)e^(-t/2) + e^(t/2)
= [(-t/2) + 1]e^(t/2)
Whoa there, cowboy. The two e’s in the top equation have different exponents (-t/2 and t/2), so you can’t factor the expression in e out, right?
I don’t need the proper solution if this is wrong; I’d just like confirmation that it IS wrong. (And if I’m wrong and it IS right as written, please 'splain me how!)
Well, what you’ve written is wrong, so, there you go. There are various plausible things which might have been intended instead, but perhaps you just want confirmation and don’t want to hear about those.
I’d be interested for my own edification, perhaps, but I don’t intend to try to rewrite the problem; that’s the author’s job. All I can do at this point is say “Please review the factoring; e has different exponents – ?” and hope they check it closely and catch my drift.
FWIW, the problem involves finding the maximum of a biological population. I’ve removed a coefficient to keep things simple, but basically
P(t) = (t)e^(-t/2)
and the equation I gave was the second step in simplifying P’(t). (I didn’t bother trying to determine whether the derivative was correct; I don’t remember enough calc for that. They did hint that the product rule and chain rule – for the e term – are needed.) But the algebra definitely looked wrong.
The exponent on the second e should also have a negative sign. So the mistake is in the first expression, and the expression after the equals sign is correct. (From another calc textbook author.)
OK, I follow the first part, but not the second. How do we factor out (e to the minus t over 2) and get a quantity times (e to the positive t over 2)?
(Me so dumb. Anything beyond “the derivative of x^n is nx^(n-1)” has long left my brain. I’d have to look up the rules for exponents, etc. It probably doesn’t help that I was tutoring a friend in pre-algebra earlier tonight, and I may be conflating some rules.)
BTW, I appreciate all the input!
Here is what I get:
d/dt[te^(-t/2)] =
td/dt[e^(-t/2)] + e^(-t/2)dt/dt (product rule*)
(-t/2)e^(-t/2) + e^(-t/2) (doing the derivatives, and using the chain rule**)
(1 - t/2))e^(-t/2) (factoring)
*(FG)’ = FG’ + F’G
** F’(G(t)) = (dF/dG)(dG/dt)
Sorry, missed the fact that the negative is missing in the last expression as well. Don’t trust textbook authors. (But, you knew that.)
No, I’m glad you chimed in! And this notation scheme stinks. As for authors, well, my job is to make sure they said what they thought they said, right? (The problem text is an insertion that may well have missed the reviewers’ pass.)
Anyway, I have enough to write an intelligent query. Thanks very much to all of you for making me look so deceptively smart. 
That’s what they did wrong.
If f = e^x then df/dx = e^x (that’s the definition of e)
Use the chain rule, as I showed above, to get the derivative of e^(-t/2).
Let g(t) = -t/2 and let f = e^g
dg/dt = -1/2
df/dg = e^g
so df/dt = -e^(-t/2)/2