The Derivative

Definition of The Derivative


The derivative of the function f(x) at the point tex2html_wrap_inline400 is given and denoted by

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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). tex2html_wrap_inline420 designate the natural logarithmic function and e the natural base for tex2html_wrap_inline420 . Recall that tex2html_wrap_inline426 .

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Chain Rule

The last formula

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

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Another similar formula is given by

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Derivative of the Inverse Function


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

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Derivative of Trigonometric Functions and their Inverses


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Recall the definitions of the trigonometric functions

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Derivative of the Exponential and Logarithmic functions


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Recall the definition of the logarithm function with base a > 0 (with tex2html_wrap_inline450 ):

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Derivative of the Hyperbolic functions and their Inverses


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Recall the definitions of the trigonometric functions

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Higher Order Derivatives


Let y = f(x). We have:

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

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Higher Derivative Formula for the Product: Leibniz Formula


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where tex2html_wrap_inline466 are the binomial coefficients. For example, we have

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