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Derivatives - Some Differentiation Rules The following...

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Unformatted text preview: Some Differentiation Rules The following pages list various rules for finding derivatives with very basic examples to show how the rules are used. The following pages are NOT formula sheets for exams or quizzes. The examples are NOT examples or samples of the problems that will be on exams. It is not a substitute for lecture or recitation. Definition: Let f ( x ) be a function. Then the derivative of f ( x ) is the function denoted f prime ( x ) given by f prime ( x ) = lim h → f ( x + h )- f ( x ) h PROVIDED this limit exists. For a particular value of x , say x = a , the derivative evaluated at x = a is given by f prime ( a ) = f prime ( x ) vextendsingle vextendsingle vextendsingle x = a = lim h → f ( a + h )- f ( a ) h PROVIDED this limit exists at x = a . A function y = f ( x ) is differentiable at a if f prime ( a ) exists, i.e., if the above limit exists. This value f prime ( a ) is called the derivative of f at x = a . Other notations: f prime ( x ) df dx d dx f D x f y prime dy dx d dx y D x y Notations for Higher Order Derivatives: 2 nd order: f primeprime ( x ) d 2 f dx 2 d 2 dx 2 f D (2) x f y primeprime d 2 y dx 2 d 2 dx 2 y D (2) x y 3 rd order: f primeprimeprime ( x ) d 3 f dx 3 d 3 dx 3 f D (3) x f y primeprimeprime d 3 y...
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