There’s a method of proving inequalities over some interval of real numbers using differentiation. For instance to prove that $x-\log(1+x) \geqslant 0$ whenever $x \geqslant 0$ we can differentiate the LHS and see that it is always positive, and since the inequality holds when $x=0$ it follows that it must also hold whenever $x\geqslant 0$ by virtue of the LHS being an increasing function.

It occurred to me that this proof resembles a “continuous proof by induction” insofar as it involved a base case $x=0$ and an “inductive step” wherein we show that if the inequality holds for $x$ then it must also hold for $x+dx$ where $dx$ is infinitesimal.

I was wondering if this method of proof can be made formal and used legitimately to prove things where other methods might fail.