The instantaneous rate of change is calculated using the limit

$latex \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}$

For many functions like polynomials, this limit may be calculated algebraically. When this limit cannot be computed algebraically or is very difficult to compute algebraically, we can use a table to estimate the limit. The problems below illustrate the table for the limit.

**Problem 1** Estimate the instantaneous rate of change of *f* with respect to *x* at *x* = 2 if

$latex \displaystyle f(x)={{x}^{\ln (x)}}$

**Solution** In this problem, *a* = 2. We need to evaluate

$latex \displaystyle \begin{matrix}

\text{Instantaneous rate of change of }f\text{ } \\

\text{with respect to }x\text{ at }x=2 \\

\end{matrix}=\underset{h\,\,\to 0}{\mathop{\lim }}\,\frac{f(2+h)-f(2)}{h}$

Since the values in the table are shown to three decimal places, we can estimate the rate to two decimal places. In the last two columns, the difference quotient rounds to 1.12 so the rate is approximately 1.12.

**Problem 2** Estimate the instantaneous rate of change of *f* with respect to *x* at *x* = 3 if

$latex \displaystyle f(x)={{x}^{\ln (x)}}$

**Solution** In this problem, *a* = 3. We need to evaluate

$latex \displaystyle \begin{matrix}

\text{Instantaneous rate of change of }f\text{ } \\

\text{with respect to }x\text{ at }x=3 \\

\end{matrix}=\underset{h\,\,\to 0}{\mathop{\lim }}\,\frac{f(3+h)-f(3)}{h}$

The table shows most values to 6 decimal places. In the last two columns, the values both round to 2.44864.