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Unformatted text preview: DERIVATIVE AS A FUNCTION If we replace a by x in the definition of the derivative of a function f at a number a , we can get f ( x ) = lim h f ( x + h )- f ( x ) h . So we can define a function that gives us the slope of the tangent line at each point, and we call it the derivative of f . Example 1. Find The derivative of the function f ( x ) = 1 x . Solution f ( x ) = lim h f ( x + h )- f ( x ) h = lim h 1 x + h- 1 x h = lim h - h ( x + h ) x h = lim h - 1 ( x + h ) x =- 1 x 2 . Lets look at the graph of f ( x ) = 1 x . If we consider the slope of the tangent line, we can see that it is negative when x > 0, but slope becomes closer to 0 as x . By looking at the graph, we can sketch the graph of the derivative, and vise versa. Example 2. The graph of a function f is given. Use this to sketch the graph of its derivative f . Historically, mathematicians used lots of different notations for the derivative....
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