The area of math that I have most enjoyed so far is Differential Calculus. When taking an early algebra class we would discover the slope of a line. I always thought that was cool at the time. However, being able to view a polynomial function on a graph and finding the slope of that line is much more gratifying.

Wikipedia of course has a very detailed explanation to Differential Calculus at http://en.wikipedia.org/wiki/Differential_calculus.

Derivatives are used to calculate the slope of a function at a given point. They can be very simple at times like when finding the derivative of $x^2+2$ the derivative is $2x$. For example, the slope of this parabola at $x=3$ is $6$. All we had to do was plug in the point at which we are wanting to calculate the slope of our function.

The most basic way to calculate a derivative is to use what is called the Power Rule.

$\frac{d}{dx} x^n=nx^{n-1}$

Derivatives can be used for many different things. As I said before they can tell you the slope, but also they are used to find the maximum or minimum of a function, and can also let you know the curvature.

Samuel CoskeyAwesome post. Derivatives tell you the rate of change of the function, which in applications could mean velocity or marginal cost.

The derivative is also a really important theoretical tool that shows up in millions of formulas and calculations—curvature being a good example of this.