How Do You Find the Instantaneous Rate of Change?

The instantaneous rate of change is calculated to find how fast one quantity changes with respect to another.

The instantaneous rate of change of  (x)with respect to x at x = a  is

\displaystyle \begin{matrix}  \text{Instantaneous rate of change of }f\text{ } \\  \text{with respect to }x\text{ at }x=a \\  \end{matrix}=\underset{h\,\,\to 0}{\mathop{\lim }}\,\frac{f(a+h)-f(a)}{h}

To apply this definition, you need to identify the point a at which the rate is to be calculated. Then the function values (a) and (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 \displaystyle f(x)=4{{x}^{2}}+2x-1 at \displaystyle x=1.

Solution Start by calculating the two function values.

m212_der_lim_1b

Once you have the function values, substitute them into the definition for instantaneous rate of change.

m212_der_lim_1a

Example 2 Find the instantaneous rate of change of \displaystyle f(x)={x}^{2}+6x at \displaystyle x=2.

Solution The function values are

m212_der_lim_2aNow put these into the limit definition of instantaneous rate of change.

m212_der_lim_2b