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Derivative_Function

# Derivative_Function - DERIVATIVE AS A FUNCTION If we...

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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 0 ( x ) = lim h 0 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 0 ( x ) = lim h 0 f ( x + h ) - f ( x ) h = lim h 0 1 x + h - 1 x h = lim h 0 - h ( x + h ) x h = lim h 0 - 1 ( x + h ) x = - 1 x 2 . Let’s 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 0 . Historically, mathematicians used lots of different notations for the derivative. For example, f 0 ( x ) = y 0 = dy dx = lim Δ x 0 Δ y Δ x = df dx = lim Δ x 0 Δ f Δ x = d dx f ( x ) = Df ( x ) and f 0 ( a ) = y 0 x = a = dy dx x = a = df dx x = a Figure 1. f ( x ) 1

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Derivative_Function - DERIVATIVE AS A FUNCTION If we...

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