Is this possible? - Controlled system equation

Question

Does this mathematical equation even make sense.

This is taken From my controlled system book, where 4.12 doesn't make sense for..

How come is that true??

Answer 1

I suppose you are confused by the $\partial$ symbol. I think in this textbook you can think of is as of simple differential $\rm d$ (not partial derivative) $$ \frac{{\rm d}T / T}{{\rm d}G / G} =\frac{\frac{{\rm d} T}{{\rm d}s} / T}{\frac{{\rm d} G}{{\rm d}s} / G} =\frac{\frac{{\rm d}\log T}{{\rm d}s}}{\frac{{\rm d} \log G}{{\rm d}s}} =\frac{{\rm d}\log T}{{\rm d} \log G} $$

Answer 2

Think of it as a shorthand notation using the Chain Rule.
Try to imagine it as the derivative of ln(f(x)).

Unless I am misunderstanding your confusion

Answer 3

It's easier to see working backwards.

Start with: $$\ln(T)$$

Take the differential: $$ \partial \ln(T)$$

Apply the chain rule: $$\frac{\partial T}{T} $$

So $$\frac{\partial T}{T} = \partial \ln(T) $$