
• In addition to defining the derivative of a function $f$ at $x = a$, the limit definition of derivative $\lim \limits _{\Delta x \to 0} \dfrac {f(a+\Delta x) - f(a)}{\Delta x}$ can be used to calculate the instantaneous rate of change of $f(x)$ with respect to $x$ at a specific value of $x$.
• Using the limit definition of derivative to compute instantaneous rates of change requires algebraically manipulating the difference quotient to cancel the $\Delta x$ in the denominator so that the limit can be easily evaluated.