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.
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
Try that using the power rule.
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:
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
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:
Let's consolidate what we just learned.
The Limit Definition of The Derivative