The instantaneous rate of change is calculated to find how fast one quantity changes with respect to another.
The instantaneous rate of change of f (x)with respect to x at x = a is
To apply this definition, you need to identify the point a at which the rate is to be calculated. Then the function values f (a) and f (a+h) are calculated and simplified. Finally, these are substituted into the limit so that it evaluated.
Example 1 Find the instantaneous rate of change of at
.
Solution Start by calculating the two function values.
Once you have the function values, substitute them into the definition for instantaneous rate of change.
Example 2 Find the instantaneous rate of change of at
.
Solution The function values are
Now put these into the limit definition of instantaneous rate of change.