Question

Is there a specific rule/formula for the derivatives and integrals for absolute values?

Answer 1

So let's say your function is g(x)=|f(x)|. Sometimes you can't differentiate at points where g(x)=0 because the graph would look "pointy" so it isn't smooth enough for you to say that the point has a gradient. For example if g(x)=|x|, you can't differentiate at x=0:

Answer 2

If you wanted to integrate the absolute value, then I would break the integral into parts and evaluate the integral for when the area would be positive and when the area would be negative:

Answer 3

For example: