Suppose two quantities \(x\) and \(y\) are related by the equation \(y = f(x)\). We say that the average rate of change of y with respect to x over the interval \([a,b]\) is

\(\dfrac{f(b)-f(a)}{b-a}\)

Suppose two quantities \(x\) and \(y\) are related by the equation \(y = f(x)\). If the limit

\(\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}\)

is defined, the we say that it is the instantaneous rate of change of y with respect to x at \(x=a\).

• Sometimes the word instantaneous is left and it is simply called as rate of change of y with respect to x at \(x=a\).
• If rate of change of y is not explicitly specified to be an average rate of change, then it is understood to be an instantaneous rate of change.
• The instantaneous rate of change of \(y\) with respect to \(x\) at \(x=a\) could also be though of as slope of the line tangent to the graph of \(f\) at the point \(P= (a,f(a))\)