Economists commonly seek to show that one variable increases in response to changes in another variable. This paper provides sufficient conditions to draw such strict monotone comparative statics conclusions in optimization problems with and without binding constraints. These results extend the lattice-theoretic results of Milgrom and Shannon [10] by imposing a stronger differential version of the single crossing property and arguing from first-order conditions. We illustrate the importance of these results through applications involving holdup problems, output decisions, optimal tax problems, and modern manufacturing.