We have known about the difference quotient for a long time now; in fact, it is often included in our tests. However, we really didn't know where it came from until we reached the last unit- Unit V! So let's take a closer look at how the difference quotient is derived.
Based on the first few concepts, we know that the difference quotient is only a part of the process when trying to find the derivative (the slope of tangent lines).
We continue to find the difference quotient by using the slope formula as shown above. After plugging in the values based on the slope formula we get [ f(x+h)-f(x)/ x+h-x]. Since the "x" in the denominator cancel out, we only have to simplify it to get [f(x+h) -f(x)/ h]....the difference quotient!!
Like I mentioned before, the difference quotient can be used to find the slope of a tangent line, which only touches the function once. We do not have to rely on the secant line because it touches the function twice. As a matter of fact, the reason why we try to find the limit as h approaches 0 is because we want "h" to be as close as possible to 0 so that both points can meet at one point instead of two like the secant line).