#
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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