The Derivative

You got it! Feels good, doesn't it?

Remember how we said that for the tangent line we had one single point? So, the slope of this line at this point is the instantaneous rate of change.

What is the Derivative, exactly?

The derivative is the slope of the line tangent to the graph of a function at a certain point.

Equivalently, the derivative of a function is the instantaneous rate of change at that point.

Of course, from high school, you remember the power rule, which we're going to re-learn later, that the derivative of

We calculated that the derivative of at the point was , so,
Try that using the power rule.
Et voila.

Derivative Notation

There are many ways of writing the derivative of a function. Newton, leibniz notation, yadda yadda yadda.

One of the most common forms is this: Sometimes y is used instead of f(x), and that's totally fair. You see the same, written as this:
But why? f(x) or y, whichever one you interpret, is a function in terms of x. So obviously, when x changes, y will change.
The derivative, when possible, will yield a function that represents the relationship between f(x) or y, and x at a certain point. So, the slope of the line tangent to the graph of the function f(x), or y, whatever.

Another way of writing it is , or , where f(x) and y represent a function in terms of x.
The little ' is read as "prime", and is yet another symbol that represents the derivative.

The last way of writing it, and the most annoying and uncommon one, is the little "dot" notation:
This, yet again, means the derivative of y.

Let's consolidate what we just learned.
The Limit Definition of The Derivative