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13142130-18-Higher-Order-Derivatives

13142130-18-Higher-Order-Derivatives - Math 135 Class Notes...

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Math 135 Business Calculus Spring 2009 Class Notes 1.8 Higher-Order Derivatives Given a function y = f ( x ), its derivative f 0 ( x ) is itself a function. This means we can take its derivative. The derivative of the derivative is called the second derivative of the function and is denoted f 00 ( x ) in Newton’s notation or d 2 y dx 2 in Leibniz’ notation We can then take the derivative of the second derivative to obtain the third derivative , denoted f 000 ( x ) in Newton’s notation or d 3 y dx 3 in Leibniz’ notation We can keep taking derivatives to obtain the fourth derivative , the fifth derivative , and so on. Past the third derivative, we usually use a superscript in parentheses instead of listing primes. f (4) ( x ) = d 4 y dx 4 f (5) ( x ) = d 5 y dx 5 . . . EXAMPLE Let f ( x ) = x 4 + 2 x 3 . Find f 0 ( x ), f 00 ( x ), f 000 ( x ), and f (4) ( x ). EXAMPLE Let y = 1 /x . Find dy/dx , d 2 y/dx 2 , and d 3 y/dx 3 . 33
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34 Chapter 1 Di ff erentiation VELOCITY AND ACCELERATION Suppose the function s ( t ) represents the distance traveled by an object as a function of time t . Then
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