BQ#7: Unit V: Deriving the Difference Quotient

Deriving the difference quotient? ..What? ...Well, si se puede!

So, first of all let's recall the difference quotient formula and remember what it has in the entire thing. 

(http://www.coolmath.com/algebra/15-functions/images/08-functions-01.gif)

Remember the song for it? Yes! It is said like this: "f of x plus h, minus f of x, divided by the letter h that's the difference quotient!" (in rhythm) 

If we take a simple, regular (x,y) graph that we are familiar with, it can be somewhat easier to  understand..to begin with. Now, like the picture below, we have a point there and instead of being 1 or 5 like a scale we would normally see, we can just call it 'x'. Then, that means that the y value would have to be f(x). If you find it confusing then think of it with numbers to make sense out of it. If we had 1 instead of x, then the y value would have to be f(1) or even if we had 5 as the x value, then the y would be f((5). Making more sense? Hopefully, so now we can continue. Like shown below, if we had the next measurement it would be x+h and the reasoning is this: if we say that the distance from x, the first point, to the second point is of h value, then our second x point would be x+h. That means that the y-value for x+h would be f(x+h). 

Almost finishing up, we would have to use the slope formula in order to continue and the formula is: 

(http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg)

The derivative is basically slope and when we are looking for the derivative, then that just means we are looking for the slope of the horizontal tangent/secant line. The two coordinates that we found we can use to plug in to the slope formula. The visual is in the image below. Okay lastly, we should clear up what in the world is a tangent and a secant line? Well, a secant line is one of which touches the graph at two point while tangent lines touch the graph at one point only. 

Still need more help and I spoke gibberish, then go ahead and watch the video below:

            (https://www.youtube.com/watch?v=XA0fZh8cXV8)

References:

(http://www.coolmath.com/algebra/15-functions/images/08-functions-01.gif)

(http://0.tqn.com/d/create/1/0/9/p/C/-/slopeformula.jpg)

(https://www.youtube.com/watch?v=XA0fZh8cXV8)