# The Derivative

## Definition of The Derivative

The derivative of the function f(x) at the point is given and denoted by

## Some Basic Derivatives

In the table below, u,v, and w are functions of the variable x. a, b, c, and n are constants (with some restrictions whenever they apply). designate the natural logarithmic function and e the natural base for . Recall that .

Chain Rule

The last formula

is known as the Chain Rule formula. It may be rewritten as

Another similar formula is given by

### Derivative of the Inverse Function

The inverse of the function y(x) is the function x(y), we have

## Derivative of Trigonometric Functions and their Inverses

Recall the definitions of the trigonometric functions

## Derivative of the Exponential and Logarithmic functions

Recall the definition of the logarithm function with base a > 0 (with ):

## Derivative of the Hyperbolic functions and their Inverses

Recall the definitions of the trigonometric functions

## Higher Order Derivatives

Let y = f(x). We have:

In some books, the following notation for higher derivatives is also used:

## Higher Derivative Formula for the Product: Leibniz Formula

where are the binomial coefficients. For example, we have

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