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A derivative of a function is a second function showing the rate of change of the dependent variable compared to the independent variable. It can be thought of as a graph of the slope of the function from which it is derived. The process of finding a derivative is called differentiation.

Definition
Let and be a function. Then the derivative of is a function defined by:
.

The derivative is undefined when this limit does not exist, that is, is not differentiable at .

## Notation

There are three types of notation typically used:

• Leibniz's notation:
• Lagrange's notation:
• Newton's notation:

Newton's notation is typically used in areas of physics, especially in differential equations. However, the notation becomes unwieldy with higher order derivatives.

## Properties

The derivative operator is linear, that is the derivative of the sum is the sum of the derivatives:

For example, using the derivative of a polynomial term:

The function can be differentiated as follows:

Applying the derivative operator multiple times gives higher order derivatives. For example, a second order derivative can be found as follows

In graphs, the derivative of a function at a number is equal to the slope of the tangent line of the graph of at the point .

Differentiation is the inverse of Integration.