Whenever we are taught thermodynamics (or classical mechanics) we are introduced to a type of transformation called Legendre transformation. When I saw that the first time I thought that it was kind of arbitrary to define the transformation in this way (for single variable functions):
$$ \frac{dg(s)}{ds}=x$$ where g is the L.T of f and s is $$\frac{df(x)}{dx}=s$$
Then I am taught in thermodynamics that sometimes it is easier to calculate the derivative of a thermodynamical variable rather than the variable itself. For example the temperature is: $$\frac{\partial U}{\partial S}=T$$ which is obviously easier to calculate than U. My question is, in the definition of L.T we imposed that the derivative of the transformation is just the inverse of the derivative of the old function (in other words, x(s)), but couldn't we just have for example defined this transformation as: $$ \frac{dg(s)}{ds}=x^2(s)$$
The only reason I can come up with is that of course the normal legendre transformation is easier to calculate but aside from that it seems as arbitrary choosing $x^2(s)$ as choosing $x(s)$ The result of the normal legendre transform as we all know is: $$g(s)=x(s)s-f(x(s))$$ if on the other hand we chose my arbitrary definition, the transformation would be: $$g(s)=sx^2(s)-2x(s)f(x(s))+2\int f(x)dx$$ , which is obviously more inconvenient.
I have a post on the physics exchange asking this question but it may belong more into this forum.
To sum up, the question is why legendre defined his transform as $ \frac{dg(s)}{ds}=x$.