The Derivative

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Definition of The Derivative

The derivative of the function f(x) at the point

is given and denoted by

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

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

Recall the definitions of the trigonometric functions

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

):

Recall the definitions of the trigonometric functions

Let y = f(x). We have:

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

where

are the binomial coefficients. For example, we have