Many people think that a better understanding of mathematics will make derivatives safer. They are wrong. Derivatives are dangerous BECAUSE OF math.
Much of higher mathematics deals with "singularities,&a... better understood by laymen as "flies in the ointment." An example comes from the fraction 1/x. It is meaningful for all x except zero. Then it "explodes"
Zero is the "singularity." It's like saying that there is there is a brook that averages three feet in depth. Except for the holes of 10 and 12 feet.
Realize that higher math is largely about "isolating," (massaging) these singularities. "The area under the curve is... If you remove X " (where the value explodes).
Math may Identify these problems but it doesn't exactly highlight them, instead, sweeping them under the rug. Derivatives have a tendency to explode--because the math behind them has the same tendency.