Lesson 14.pdf - 14 Derivatives Derivative of a function function Definition Let I \u2282 R be an interval and let f I \u2192 R be a 1 The function f is

Lesson 14.pdf - 14 Derivatives Derivative of a function...

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14. Derivatives July 26, 2019 Derivative of a function Definition Let I R be an interval, and let f : I R be a function. 1. The function f is differentiable at c I if the limit lim x c f ( x ) - f ( c ) x - c (1) exists. 2. The limit is called the derivative of f at c and is denoted by f 0 ( c ). 3. If f is differentiable at each point x in a set S I , then f is differentiable on S , and the function f 0 : S R is called the derivative of f . [Examples and remark] 1. Another version of the limit (1): lim h 0 f ( c + h ) - f ( c ) h . 2. Let f ( x ) = k be constant. For any c R , f ( c + h ) - f ( c ) h = k - k h = 0 h = 0 . Then f 0 ( c ) = lim h 0 f ( c + h ) - f ( c ) h = 0. 116
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