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

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

$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}$

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

**Solution** Start by calculating the two function values.

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

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

**Solution** The function values are

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