That's so Derivative

We know that the slope, , is the rate of change of with respect to .  This was easy to see on a line, but on a curve, it's not so straight forward.  For instance, take the equation graphed as:

The rate of change...changes.  When x is less than 0, y gets smaller as x gets larger.  Conversely, when x is larger than 0, y gets larger as x gets larger.  This means we can't calculate the slope of the curve as a whole.  We can do two things however that are just as helpful.  We can calculate the slope of the curve at a point, and we can create a formula representing the slope of the curve at a point.

The rate of change of a curve at a specific point is known as the derivative.  The formal definition of a derivative refers to the slope of the tangent line of a curve.  Mysterious?  Maybe.  Difficult to understand? Not really.  The tangent line is very simple actually.  It's simply a line that touches a curve at a single point without passing through that point.  As a result, the tangent line is the best linear representation of a curve at a single point and thus, finding the slope of the tangent line gives you the slope of the curve at that point.  Here's what the tangent line looks like for our graph at the point :

The line just skims by the curve.  Gently caressing our point while avoiding surrounding points..  Without it we are lost in a slopeless void.  It's poetic really. To honor the tangent line, I present to you this haiku:

Gentle tangent line
So simple in your beauty
Thank you for your slope
 

Our goal then is to find the slope of the tangent line.  The first step oddly enough is to find the tangent line.
We'll start with the equation from above: .  For this example, we're going to use the point where .  When , we know that as well, so our point will be .We'll start by comparing our representative point, , and other points on the graph.  For demonstration, we'll use the following set of values for x: .  Using those values, we get the following ordered pairs of coordinates represented as :

• 
• 
• 

We can draw a line between each of these points and our representative point and get the following graph:

Each of these lines pass through exactly two points on the graph and are by definition, secant lines.  These secant lines aren't helpful on their own, but with a little thought, we can intuit that the tangent line is actually the limit of these secant lines.  That's a bit of a tricky concept, but it's one that is ultimately simple and rewarding when it finally clicks.

Think of it this way, as the points on the line get closer and closer to the value of our representative point, the secant lines get closer and closer to running parallel to the tangent line.

It's time then to generalize an equation for our secant lines.  The secant line takes two points.  The first of the points is .  We can make that more specific though because we're math people.  In actuality, is equal to .  Now our second point has a value for x that is an arbitrary distance from our first point.  We'll call this distance and show our second point as .

Since we know that the slope is we can represent the slope of our secant line as which is equal to

Now, we move back to the tangent line concept.  We've already agreed that the secant lines get closer to the tangent line as the second point in the line gets closer to the first point.  The ultimate goal then is for there to be no distance between the first and second point.  This is hard to calculate geometrically, but it means we can use our already garnered knowledge of limits to fill in the gap.

So our goal is to find the value for the slope of the secant line as the value for approaches 0.  This means that the equation for the slope of the tangent line can be represented:

That's it.  The above equation will show you the slope of a line tangent to the curve.  That's a derivative.  Pretty cool stuff.  All that's left is to plug in our equation, , and find the limit.

The derivative of our equation is therefor:

Which equals:

And simplifies to:

And further simplifies to:

Calculating our limit, we can conclude that the derivative of the function is equal to .  This is also known as the first derivative of and can be represented in multiple ways.  I will most likely stick with either or , both of which mean the same thing.

So, when , .  Using this new equation, we can find the slope of a curve at a specific point.  Using our representative point from above, we can find the slope as follows:

That my friends, is the basis of differentiation.  Enjoy.