Question

What is the relation between limits, derivatives and instantaneous rate of change?

Answer 1

Both derivatives and instantaneous rates of change are defined as limits.

Explanation:

Depending on how we are interpreting the difference quotient we get either a derivative, the slope of a tangent line or an instantaneous rate of change.

A derivative is defined to be a limit. It is the limit as `h rarr 0` of the difference quotient `(f(x+h)-f(x))/h`

The instantaneous rate of change is also a limit. It is a limit of an average rate of change. Because the average rate of change is expressed as `(f(x+h)-f(x))/h`, the instantaneous rate of change is also a limit of the difference quotient.