Example 1

Power functions:
We consider the function $y(x) = x^3 + 2x^2 - 3$ and wish to determine the average change in the function value if $x$ increases from $x=2$ to $x=5$. This implies that
$$ \Delta x = 5-2=3.$$
We also need the values $y(x)=y(2)$ and $y(x+\Delta x) = y(5)$:
$$\begin{align}
y(2) &= 2^3 + 2\cdot2^2 - 3 = 13,\\
y(5) &= 5^3 + 2\cdot5^2 - 3 = 172.
\end{align}$$
Now we are able to determine the difference quotient:
$$ \dfrac{\Delta y}{\Delta x} = \dfrac{y(x+\Delta x)-y(x)}{\Delta x} = \dfrac{y(5)-y(2)}{3} = \dfrac{172-13}{3} = \dfrac{159}{3} = 53.$$

Natural logarithm:
We consider another function, $f(x) = \ln(x^2 + 1)$, and wish to determine the average change in the function value if $x$ decreases from $x=2$ to $x=0$. We deal with
$$\Delta x = 0 - 2 = -2.$$
We also need the values $f(x)=f(2)$ and $f(x+\Delta x) = f(0)$:
$$\begin{align}
f(2) &= \ln(2^2 + 1) = \ln(5),\\
f(0) &= \ln(0^2 + 1) = \ln(1) = 0.
\end{align}$$
Now we are able to determine the difference quotient:
$$ \dfrac{\Delta f}{\Delta x} = \dfrac{f(x+\Delta x)-f(x)}{\Delta x} = \dfrac{f(0)-f(2)}{-2} = \dfrac{0-\ln(5)}{-2} = \tfrac{1}{2}\ln(5).$$