Problems with thermodynamic derivatives
It's a consequence of the triple product rule for partial derivatives of thermodynamic state variables.

In summary, if you have three interdependent (not independent) variables, then this is true:

$$
\left ( \frac{\partial{X}}{\partial{Y}} \right )_Z \left ( \frac{\partial{Z}}{\partial{X}} \right )_Y = - \left ( \frac{\partial{Z}}{\partial{Y}} \right )_X
$$

Substituting in your variables, if we let $X = T$, $Y = P$, and $Z = S$, we get:

$$
- \left ( \frac{\partial{S}}{\partial{T}} \right )_P \left ( \frac{\partial{T}}{\partial{P}} \right )_S =  \left ( \frac{\partial{S}}{\partial{P}} \right )_T
$$

Note that the order of multiplication of the two partial derivatives doesn't matter.

Now, why this rule holds for thermodynamic variables is a little bit more complicated - but I think the wikipedia article that I linked to has a pretty good derivation if you are interested.