Therefore,\begin{align}\frac{dT_i}{T_i} &= 3.73\frac{dr_s}{r_s} \end{align}

[Je m'appelle]

Homework Statement

In a few words, show that

$$\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}$$

Where,

$$r_s = \lim_{P_i\rightarrow 0}\frac{P_s}{P_i}$$

There's a picture below for more details.

Homework Equations

$$T_i = \frac{100}{r_s - 1}$$

The Attempt at a Solution

So I tried this,

$$T_i = \frac{100}{r_s - 1}$$

Applying 'ln' to both sides,

$$ln(T_i) = ln(\frac{100}{r_s - 1})$$

And then deriving both sides by $$\frac{d}{dr_s}$$, as $$T_i$$ is a function of $$r_s$$ (right?) I get to

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{r_s - 1}{100}\frac{-100}{(r_s - 1)^2}\frac{dr_s}{dr_s}$$

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{-1}{r_s - 1}$$

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{1}{1 - r_s}$$

$$\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s}$$

Where

$$r_s = \lim_{P_i\rightarrow 0} (\frac{P_s}{P_i})$$But this looks completely nonsense when comparing to what I was asked to prove in the first place.

I should have gotten to something like

$$\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}$$

But instead I got to

$$\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s}$$

Thoughts?

 

[jbsteele]

A:Let's solve it in a different way.\begin{align}T_i &= \frac{100}{r_s - 1} \\\frac{dT_i}{dr_s} &= \frac{-100}{(r_s - 1)^2}\\\frac{dT_i}{T_i} &= \frac{dT_i}{dr_s}\frac{dr_s}{T_i} = \frac{-100}{(r_s - 1)^2}\frac{dr_s}{\frac{100}{r_s - 1}}\\&= -(r_s - 1)\frac{dr_s}{r_s}\\&= -3.73\frac{dr_s}{r_s}\end{align}So,\begin{align}\frac{dT_i}{T_i} &= -3.73\frac{dr_s}{r_s}\\\frac{dT_i}{T_i} + 3.73\frac{dr_s}{r_s} &= 0\end{align}