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The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a.

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Presentation on theme: "The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a."— Presentation transcript:

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2 The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a function of x.

3 f’(x) “f prime of x” dy “the derivative of y with respect to x” dx “ dy – dx” y’ “y prime” d [f(x)] dx Dx[y]

4 dy = lim ∆y dx ∆x 0 ∆x = lim f(x + ∆x) – f(x) ∆x 0 ∆x = f’(x)

5 Alternative Form of a Derivative f’(c) = lim f(x) - f(c) x c x - c

6 The existence of the limit requires that the one-sided limits exist and are equal. f’(c) = lim f(x) - f(c) x c - x - c f’(c) = lim f(x) - f(c) x c + x - c (The derivative from the left) (The derivative from the right)

7 f is differentiable on the closed interval [a, b] if it is differentiable on (a, b) and if the derivative from the right at a and the derivative from the left at b both exist.

8 Example: f(x) = |x – 2|

9 Conditions where a function is not differentiable: 1. At a point at which a graph has a sharp turn. 2. At a point at which a graph has a vertical tangent line. 3. At a point at which the function is not continuous.

10 Theorem: Differentiability Implies Continuity If “f” is differentiable at x = c, then f is continuous at x = c.


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