Expert: Sherry Wallin - 12/12/2010

Question
QUESTION: Recently, I've been fooling around with my ti-89 to figure out why x^(1/x) maximizes at e. I used derivitives, but it the process, saw something that didn't make sense.
If I take the derivitive of t^(1/x) for x, I get (-t^(1/x)*ln(t))/x^2) However, when I plug in x for (t), I get ((1/(x^2))-((ln(x))/(x^2)))*(x^(1/x)). This doesn't happen when I plug in a numerical value or non-x variable, so why does it happen when I plug in x. P.S. something similar happens with the derivitive of(t^x).

ANSWER: Hi Terence~

I am trying to follow your argument and you did say you are taking the derivative of t^(1/x) with respect to x. I get the same expression when t is in the function but not the same when substituting x. I believe this is what you should have:
-(lnx)*[x^(1/x)/x^2] = -(lnx)/x^(2-(1/x)). See if this fixes the problem you are having.

Math Prof

(t^x)' = ln(t)*t^x(x') = ln(t)*t^x

---------- FOLLOW-UP ----------

QUESTION: Okay, I see what you're saying, but why doesn't it equal
(-(lnx)*x^(1/x))/(x^2)?


ANSWER: Terence~

Why doesn't what equal what? It is not clear what you are inquiring about. If you are asking why -(lnx)*[x^(1/x)/x^2] = -(lnx)/x^(2-(1/x))
then all I am doing to combining factors of x: x^(1/x)/x^2 = x((1/x)-2) which when you move it downstairs becomes 1/(x^(2-(1/x)). If this isn't addressing what you are asking please send again with more detail and try not to use words like it, specify what it is please.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Sorry I wasn't clear.
If you take the derivitive of t^(1/x,x you get a certain expression. My question is why is it that plugging x in for t (x^(1/x)), I don't get an expression that is the samething but with t in the place of x. Why is it that I get a completly different expression?


Answer
Terrence~

I still am not sure what you are trying to find out. Perhaps you have typed it wrong? If what you want to do is substitute x where t is of course you are not going to get the same expression because you are using x in two different places. Try this instead, take the derivative of x in place of t and then see what you get, maybe that will make it clearer.

Math Prof