What is difference quotient used for in real life?

What they're going to do is use that formula, then let h get very small ... so small, in fact, that goes to zero "in the limit." And when you do that, you're doing elementary calculus.

Here's an example. Let's take a simple quadratic like y = x^2. This is a parabola that opens upward, but it's curving all the way. It passes through the point (2, 4), and the question is to find the slope of the line tangent to the curve at that point. (The tangent tells you the direction that the curve is pointed. This direction, of course, keeps changing.)

So we have y = f(x) = x^2, and we have your formula [f(x+h) - f(x)] / h.

[f(x+h) - f(x)] / h = [(x+h)^2 - x^2] / h = [(x^2 + 2hx + h^2) - x^2] / h = (2hx + h^2) / h = 2x + h

h is a very small number -- something like h = 0.0000000001. It's so small that you can let it go to zero without losing much, something like this, at x=2 from above:

[f(x+h) - f(x)] / h = 2x + h = 2 • 2 + 0.0000000001 = 4.0000000001 ==> 4

What you've shown is that the slope of y = x^2 at any point (x, y) is 2x. At the point (2, 4), the slope of the curve is 4.

That's elementary calculus, and it all starts with your difference quotient formula.

Does this help?