Question

How do you find the instantaneous rate of change of f at `x=a`?

Answer 1

By definition, the instantaneous rate of change of `f` at `x=a` is the limit of average rates of change.

The average rate of change can be written in different ways.

Because of this, the instantaneous rate of change of `f` at `a` can be written as either:

`lim_(xrarra)(f(x)-f(a))/(x-a)`

or

`lim_(hrarr0)(f(a+h)-f(a))/h`

In either case, attempting to find the limit by substitution lead to the indeterminate form `0/0`. So techniques need to be developed to evaluate these limits. The techniques used depend on the function `f`.