Real world example of using difference quotient vs. derivative
You can see the applications in microeconomics. A tiny example could be on elasticity of demand. 

Let's say 

$$ D = \frac{tC}{P+t} $$

where $P, t, C$ are respectively price, tax and constant. If you like to see the effect of tax change on demand you can use the difference quotient ;

$$\Delta D = \frac{(t + \Delta t C)}{P + t + \Delta t} - \frac{tC}{P + t}$$

I skip the intermediary calculations. This one yields ;

$$\frac{\Delta P}{\Delta t} = \frac{P.C}{(P+t)^{2}+\Delta t.(P+t)}$$

Hope that it helps.