What happens if we compute this average rate of change of {{MATH_36}} for each value of {{MATH_37}} as {{MATH_38}} gets closer and closer to {{MATH_39}}? In other words, what is the meaning of the limit {{MATH_40}} provided that the limit exists? Naturally, we call this limit the {{CMD_41}}instantaneous rate of change{{CMD_42}} of the function {{MATH_43}} at {{MATH_44}}. {{COMMENT_45}} {{COMMENT_46}} {{CMD_47}} (In this example we will use abreviation AvRCh for the average rate of change, and InRCh for the instantaneous rate of change.) Let {{MATH_48}}. {{CMD_49}} {{CMD_50}} Find the expression for the average rate of change of {{MATH_51}} between the points {{MATH_52}} and {{MATH_53}}. {{MATH_54}} Now evaluate the function, {{MATH_55}} Simplify, {{MATH_56}} {{CMD_57}} Find the average rate of change of {{MATH_58}} between the points {{MATH_59}} and {{MATH_60}}, {{MATH_61}}. {{MATH_62}} Now substitute in for the function, {{MATH_63}} Simplify the top, {{MATH_64}} Factor, {{MATH_65}} Factor and cancel, {{MATH_66}} {{CMD_67}} Find the instantaneous rate of change of {{MATH_68}} at the point {{MATH_69}}. {{MATH_70}} So, the instantaneous rate of change is the limit, as {{MATH_71}}, of average rates of change of {{MATH_72}} between points {{MATH_73}} and {{MATH_74}}, for {{MATH_75}}. We have already computed an expression for the average rate of change for all {{MATH_76}}. Therefore, {{MATH_77}} {{CMD_78}} {{CMD_79}}