Review - Introduction to Differential Calculus

Linear equations have form , this is known as the slope-intercept form
m is the slope of the line, calculated by
c represents the y-intercept of the graph, the value of y for which x = 0.
Linear equations are also expressed in the form , where represents a point this line must pass through. We can calculate c from this.
The slope of a line represents its direction, it's a mathematical way of saying "for every 1 change in x, y changes by this much".
The slope of the secant line is taken over two points of another graph and represents the average rate of change of the function over that interval.

When we make the distance of that interval, x, infinitesimally small, we go from an interval to a single point, getting the slope of the line tangent to the graph of that point.
The slope of this line tangent to the graph of the function at that point represents the instantaneous rate of change of that function at that point.
This slope is also known as The Derivative of the function at that point.

The derivative of a function at a point, written as , , , , etc... if it exists, is the slope of the line tangent to the graph at that point.
This is known as the slope of the secant line over an infinitesimally small distance as the limit definition of the derivative:
If that limit exists at a point, the function is differentiable at that point.
Because of the nature of limits, for a function to be differentiable, it has to be continuous. If we can prove discontinuity, we can prove it's not differentiable.
If a function is differentiable, it is continuous. But the inverse is NOT TRUE. A function that is continuous is not necessarily differentiable.
Any derivatives of a continuous differentiable function are also continuous over their domains.

We want to have non-negative inputs for all the x axis. We will define the piecewise function:

We are approaching x = 3 from the left as

We are approaching x = 3 from the right as

The function is not differentiable at x = 3