A derivative is an operation on a function which describes the rate of change of a single variable function with respect to its input.

Notation

There are 2 main notations for the derivative each of which have their own strengths

• Leibniz's notation
• Lagrange's notation

Definition

The derivative can formally be defined for a given function using the following limit .

^formal-derivative-definition

Origin

This limit comes from the notion that the slope of a function from to can be calculated using rise over run as follows

Using a limit we can describe what happens as approaches 0 (in the formal definition this is set the be equal to ). As the difference between points approaches 0 we get the instantaneous rate of change at .

Re-writing the previous equation with we get the following equation

We can then apply the limit as approaches 0 as in the original definition giving us the instantaneous rate of change of the function .

Calculating

Derivatives can be calculated using the formal limit defined above, graphically by estimating the tangent line with a secant line between two points very near to the desired derivative location on the graph, or by using special differentiation rules. Usually the fastest method for calculating derivatives when possible is using differentiation rules.