Slope of a Parabola

Slope

The slope of a line is just the rise over the run. If I plot the line as a graph of y vs. x, then the slope is the change of y divided by the change of x.
Let's represent the change in y by the symbol dy, and the change in x by the symbol dx. Then the slope is just dy/dx.
Now how do I compute the slope of a function at a point? Well, you compute the ratio of dy/dx at a point on the curve when dx gets infinitesimally small.

Deriving the Slope of a Parabola

Start with the equation of a parabola: y(x) = a x²
We will look at two points, one at x and one very close by, at x+dx. Then we will compute the values of y at x and at x+dx. The difference between the two y values is dy.
So at x we have y(x)=a x². And at x+dx we have y(x+dx)= a (x+dx)² = a x² + 2a x dx + a dx²
So dy = y(x+dx) - y(x) = (a x² + 2a x dx + a dx²) - (a x²) = 2a x dx + a dx²

Dividing dy by dx gives us our slope:
Slope = dy/dx = (2a x dx + a dx²) / dx = 2ax + a dx

Now we make dx go infinitesimally small, like, to zero. This gives us the slope at the point:
Slope = 2ax

It seems like a trick, to introduce dx in the calculations, then to make it go to zero, but it really works.