Let's start with the average rate of change of the function {{MATH_11}} as the input changes from {{MATH_12}} to {{MATH_13}}. We will introduce a new variable, {{MATH_14}}, to denote the difference between {{MATH_15}} and {{MATH_16}}. That is {{MATH_17}} or {{MATH_18}}. Take a look at the figure below. {{CMD_19}} {{ENV_20}} {{CMD_21}} Now we can write {{MATH_22}} What happens if {{MATH_23}}? In other words, what is the meaning of the limit {{MATH_24}} This limit represents {{MATH_25}}, the instantaneous rate of change of {{MATH_26}} at {{MATH_27}}! Therefore, we have an alternate way of writing the definition of the derivative at the point {{MATH_28}}, namely {{MATH_29}}