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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 .

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derivative
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
Note on Notations
  • dx does not mean “d times x!!”
  • dy does not mean “d times y!!”
example
Example
  • Find the derivative of the function f(x) = x3.
slide6
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.
example1
Example
  • Find the derivative of

(Multiply by the conjugate)

a note from the example
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
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
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.
example2
Example
  • Show that the following function has a left- and right-hand derivatives at x = 0, but no derivative there.