Chain rule / Taylor expansion / functional derivative

[binbagsss]

Homework Statement

To show that $\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0$
where $p=p(z)$, $p'=p(z+dz)$, $s'=s(z+dz)$, $s=s(z)$

Homework Equations

I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions, but if someone could please give me a clue where to start.

The Attempt at a Solution

as above.

many thanks in advance.

 

[Charles Link]

Suggest a Taylor expansion of $\rho(p',s')$ in just the s variable about s. One comment is since s and s' differ by a differential amount, the the terms of the first inequality differ by only a differential amount. I think the inequality you are trying to prove actually needs a "dz" tacked on to (multiplying) the product of the derivatives. I would be interested in seeing if other readers concur. "dz" does not need to be positive.