Derivative Operator

The derivative of a function with a respect to returns a function that represents the change in over the change in . For example, the derivative of the line represented by the function is equal the constant function .

Notation

The derivative of a function is denoted multiple ways. For all of these examples, we are expressing the derivative of the function with respect to the variable . The first way to denote the derivative is shown below.

In plain language, this expressions means the change in over the change in and represents the derivative of the function with respect to . The second way to denote the derivative is a single quote more after the function name and before the function’s arguments.

In plain language, this expression represents the first derivative of the function with respect to the variable . The third notation is Leibniz’s notation.

In plain language, this means take the derivative of the expression in brackets with respect to the variable . In this case, the expression is the function . The variable in the expression indicates which variable the derivative is being taken with respect to.

Limit Definition

The definition of the derivative of a function is given in terms of the limit as the change in goes to . The change in is denoted with the (capital delta) symbol as . The definition is given below.

Examples

The table below lists some common derivatives. Note, the constant coeffecients are highlighted in red so they are not confused with the variable .

Function Derivative
Constant
Line
Quadratic
Cubic
Polynomial
Sine
Cosine
Exponential
Logarithm

Properties

Often when differentiating a function the properties of the derivative operator are used to transform the function.

Property Equation
Product Rule
Quotient Rule
Chain Rule

Leibniz Notation

Property Equation
Product Rule (Leibniz)
Quotient Rule (Leibniz)
Chain Rule