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Warm Up. Find the slope of the tangent line to at x=2. Answer: m= -4. Derivative of a Function. 3.1. Goal. I will be able understand the relationship between a function and its derivative as well as recognize when a function will not be differentiable. New calendar . Definition.

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Presentation Transcript
warm up
Warm Up
  • Find the slope of the tangent line to

at x=2.

  • Answer: m= -4
slide3
Goal
  • I will be able understand the relationship between a function and its derivative as well as recognize when a function will not be differentiable.
  • New calendar
definition
Definition
  • The derivative is the formula for slope of a tangent line, instantaneous speed, or velocity of an object.
alternate formula
Alternate Formula
  • If asked to find the derivative at a point x=a.
notation
Notation
  • There are many ways to denote the derivative. They can all be found at the top of page 101.
  • I will also give them to you now…
slide8

“the derivative of f with respect to x”

Y prime

“the derivative of y with respect to x”

“the derivative of f with respect to x”

“the derivative of f of x”

slide9

Note:

dx does not mean d times x !

dy does not mean d times y !

slide10

does not mean !

does not mean !

Note:

(except when it is convenient to think of it as division.)

(except when it is convenient to think of it as division.)

example
Example
  • Use the definition to find the derivative of at a=1.
  • Answer:
graphing f x
Graphing f’(x)
  • To graph the derivative, estimate the slope at a few points, then plot those values on the new graph.
when can you not find the derivative
When can you not find the derivative?
  • Differentiability implies continuity! If a function is not continuous, it is not differentiable.
  • Cusps/points, vertical asymptotes, jumps/gaps