2.2 The Derivative function & 2.3 Interpretations of the derivative

Main Points:
The derivative function is defined as the Instantaneous rate of change of f at x. The derivative function is positive when the slope of the original function is positive. The opposite is true when the original function has a negative slope. The sign of f ' corresponds with the direction of the slope of f (increasing, decreasing, or constant). Finding the slope at a given point from numerical values given is one way of finding the derivative. To get a more accurate result from this method average the slope from the values on either side of the target point. To find the derivative from a formula, you can take two very close values, plug them into the equation, find their difference and divide that by the difference between the two x values. The derivative can also be described as (delta y/delta x). This can be written (dy/dx). The derivative can be written to represent values such as velocity (v=ds(position)/dt). A local linear approximation can be made when the derivative is given. Delta y approximately equals f '(x) * delta x.

Challenges:
I was confused as to why Leibniz's notation is difficult to use to express what f '(x) equals. The whole infitesimally small change in x and y way of describing derivative was also a bit confounding. I just thought it was a poor way of explaining it.

Reflections:
I never got this extensive background of how a derivative is essentially derived, so it is interesting to see all of the very painfull methods of finding something so simple. In my calculus class last year, the teacher didn't bother explaining the how or why part of calculus, we just did it, so I appreciate this........kind of.