Brian
Derivative
How one thing changes based on a change in another thing. For example, acceleration is the derivative of velocity. How the velocity changes with respect to time is acceleration.
Integral
Summing up the values of a dependent variable over a range of an independent variable.
Example
Assume that 10 grams of water flows out of a faucet every second and the faucet is kept running for 10 seconds.
Derivative: There are 10 grams of water flowing out of the faucet every second so the derivative, or rate of change of water mass with respect to time, is 10 mg/s.
Integral over 6 seconds: If you sum up 10 mg/s over 6 seconds you get 60 mg, which is the amount of water accumulated in the sink after 6 seconds.
