The way that Calculus is traditionally taught gives a false impression that every function worth looking at can be differentiated using the rules of differentiation.  This comes from a misconception that any function worth looking at can be described by an algebraic formula, or using trigonometric or logarithmic functions.

That's just not the case: the most common everyday functions don't have any formulas.  Some examples:

 1. Price of a company stock over several decades.
 2. Volume of water in a water tower over the course of a week
 3. Median price of a house in your area (adjusted for inflation), over the course of 100 years.
 4. US National Debt over the last two hundred years.
 5. US Deficit

For such functions, rate of change has a very real meaning.  I find that students who had Calculus in high-school are stumped if I give them an example like that and ask them to graph the rate at which, say, the US national debt has changed throughout US history, and how that relates to the deficit.  

Understanding the derivative as both rate of change and the slope of the tangent line helps, and the only good way to tie those concepts is with using limits.