Inequalities and Differentiation Having become so accustomed to differentiation and integration being applied just like normal algebraic operators, I was somewhat suprised yesterday when I realized that $f(x) \geq g(x)$ does not imply $f'(x) \geq g'(x)$. Intuitively this makes sense, but it's somewhat surprising considering that $f(x) \geq g(x) \Rightarrow \int_a^b f(x) \geq \int_a^b g(x)$. Can anyone think of a way to demonstrate this 'symmetry breaking' with some rigor? Thanks in advance!

We usually say that _inequalities can be integrated, but they cannot be differentiated_. Which is not surprising, since integration is essentially summing, and summing preserves inequalities. Differentiation, on the other hand, is more like subtracting, that does _not_ preserve inequalities.

The following toy model might be useful. Suppose that $$ a(1) \le b(1), \quad a(2)\le b(2). $$ You can infer that $$ a(1)+a(2)\le b(1)+b(2). $$ But you _cannot_ infer that $$a(2)-a(1)\le b(2)-b(1).$$ Try with $a(1)=0, a(2)=1, b(1)=2, b(2)=2$.