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The derivative of a function is the function that returns the rate of change at any input point.

The Definition of the Derivative
"The derivative of a function is also a function. The resulting function can be evaluated at any input to determine the slope for a small range around the input, commonly 0.001 either way."

Mathematical Definition
"The mathematical definition for the derivative of a function $f(x)$ is the limit of the slope of the line going through two incredibly close points:"

$${df \over dx} = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$Representation and Notation
There are 3 commonly used notations for the derivative, each named after a famous mathematician:


 * 1) Gottfried Wilhelm Leibniz's notation, the one used above
 * 2) Joseph Louis Lagrange's notation, and
 * 3) Newton's Notation