3.1 Derivatives

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# 3.1 Derivatives - PowerPoint PPT Presentation

3.1 Derivatives. 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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### 3.1 Derivatives

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.
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) = x3.
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)

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 graph of y’ look like?

Remember: y’ is the slope of y.

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 = 0, but no derivative there.