3 1 Derivatives Derivative A derivative of a

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3. 1 Derivatives

3. 1 Derivatives

Derivative �A derivative of a function is the instantaneous rate of change of the

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.

Symbols Used to Denote Derivatives

Symbols Used to Denote Derivatives

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

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

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

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

Note �Your book talks about an “alternate definition. ” Do not worry about using

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.

Example �Find the derivative of (Multiply by the conjugate)

Example �Find the derivative of (Multiply by the conjugate)

A Note from the Example �From the previous example: �What was the domain of

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.

Functions and Derivatives Graphically The function f(x) has the following graph: What does the

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.

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

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

One-Sided Derivatives �Since derivatives involve limits, in order for a derivative to exist at

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.

Example �Show that the following function has a left- and right-hand derivatives at x

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

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