Question

Hi, I got a question to Exercise 2E-10. Along the way to the solution T'=1/t, a T somehow magically disappears. Thus, I rather obtain the diff. eq. T'=T/t as a solution.
Is there an error or does anybody have an explanation?

Answer 1

First of all T´=-1/t not just 1/t. Secondly when you obtain a derivative of an explicite function definition the function definition always disappear and is replaced by it's derivative.

So if y=sin(x) then y´=cos(x). Notice that y disapperas and is replaced by the derivate.

Thirdly the excercise asked what T´(t) (the derivate of T) is, not what T(t) is.

Answer 2

Hi Topi, thanks for answering! I'm not quite shure you got me right, I'm refering to this line here:
\[\left( r ^{2} T \right)\prime = 2rr \prime T + r ^{2}T \prime = 0\]
from which T' is obtained as
\[T \prime = - 2 r \prime/r\]
But exactly here I am missing the T.

Answer 3

Ok. Didn't notice that.

You are correct that there is a discrepancy there, but I'm not sure how to handle that. Except that you cand substitute the equation of T in there and solve then for T'.

Answer 4

I also had trouble with the answer given in the answer key - I came to the same conclusion. If I keep the capital T in the derivation, i get answer: -T/t instead of the proposed -1/t .

Answer 5

You cannot just keep the T in the derivation as a constant as it's a function of t.