Find some shortcuts for taking derivatives. For example, if the function is of the formy=c, wherecis a constant, then you know the graph is a horizontal line, with a slope of zero, and you don't have to even go through the limit gyrations to get the derivative.

Do some experimenting with the limit definition on some easy functions to see what you get. When you post a shortcut, you must also post a good portion of the algebraic (or geometric) argument that supports it. No numerical or graphical (calculator) approximations will count as justification you need to argue with algebra or geometry.