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3.1 Derivatives

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Derivative A derivative of a function is the instantaneous rate of change of the function at any point in its domain. We say this is the derivative of f with respect to the variable x. If this limit exists, then the function is differentiable.

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Symbols Used to Denote Derivatives

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Note on Notations dx does not mean “d times x!!” dy does not mean “d times y!!”

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Example Find the derivative of the function f(x) = x 3.

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Note Your book talks about an “alternate definition.” Do not worry about using the “alternate definition.” You will never see it on an AP exam! If the directions on your HW say to use the alternate definition, use the regular definition of the derivative.

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Example Find the derivative of (Multiply by the conjugate)

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A Note from the Example From the previous example: What was the domain of f? [0, ∞) What was the domain of f’? (0, ∞) Significance??? Sometimes the domain of the derivative of a function may be smaller than the domain of the function.

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Functions and Derivatives Graphically The function f(x) has the following graph: What does the graph of y’ look like? Remember: y’ is the slope of y.

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Functions/Derivatives Graphically The derivative is defined at the end points of a function on a closed interval.

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One-Sided Derivatives Since derivatives involve limits, in order for a derivative to exist at a certain point, its derivative from the left has to equal its derivative from the right.

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Example Show that the following function has a left- and right-hand derivatives at x = 0, but no derivative there.

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CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a))

CHAPTER 2 2.4 Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a))

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