Lots of art therapy these days.

Definition of the derivative. How do we find an expression for the exact slope at a single point, when the slope equation requires two points? 1/9

Start by defining two generic points on a function f(x), separated by some distance delta x. 2/9

Then we can find the expression for the slope between them. 3/9

That is... 4/9

But we don't want the average slope between two points on the function. We want the *exact* slope at a single point. 5/9

Brainstorm montage (insert music)... 6/9

As the two points get really close together (delta x is very small), the closer the slope is to representing the slope at a single point. 7/9

And as the distance between the two points gets infinitely small (delta x zero), the slope between them is the exact slope at a single point. 8/9

Which gets us the definition of the derivative - the expression for instantaneous slope at any point on function f(x), found by finding the slope between two generic points on the function as the distance between them gets infinitely small.

End.
9/9