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Unformatted text preview: Section 2.8: Derivatives Instructor: Ms. Hoa Nguyen (email@example.com) Derivatives The derivative of a function f at a number a is: f ( a ) = lim h f ( a + h )- f ( a ) h if this limit exists. Let x = a + h h = x- a , then: f ( a ) = lim x a f ( x )- f ( a ) x- a if this limit exists. The function f is called the derivative of f because it has been derived from f by the limiting operation in the above equations. Refer to Section 2.7 to understand the interpretation of the derivatives as a slope of a tangent or an instantaneous rate of change . Example 1 of Section 2.8 (textbook): The graphs of a function f ( x ) and its derivative f ( x ): f ( x ) > 0 on ( a,b ) f increases on ( a,b ). f ( x ) < 0 on ( a,b ) f decreases on ( a,b ). f ( x ) = 0 at x = c the tangent at the point ( c,f ( c )) is horizontal. Questions : Is the derivative a function? Recall that a function is a map which associates one number to another....
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