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THE CHAIN RULE

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Find the derivative of a trigonometric function using the Chain Rule. THE CHAIN RULE ... TRIGONOMETRIC FUNCTIONS AND THE CHAIN RULE ... – PowerPoint PPT presentation

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Title: THE CHAIN RULE


1
THE CHAIN RULE
  • Section 2.4

2
When you are done with your homework, you should
be able to
  • Find the derivative of a composite function using
    the Chain Rule
  • Find the derivative of a function using the
    General Power Rule
  • Simplify the derivative of a function using
    algebra
  • Find the derivative of a trigonometric function
    using the Chain Rule

3
THE CHAIN RULE
  • If is a differentiable function of u
    and is a differentiable function of
    x, then is a differentiable
    function of x and

4
PROOF
  • Let . Then using the
    alternative form of the derivative, you need to
    show that for
  • . We need
    to think about the behavior of g as x approaches
    c. We will assume that for
    values of x other than c. In Appendix A, the
    author shows how to use the differentiability of
    f and g to address the situation when there are
    other values of x, other than c such that
    . Since g is differentiable, it is also
    continuous, and it follows that
    .

5
PROOF CONTINUED
6
Find the derivative of
  • None of the above

7
THE GENERAL POWER RULE
  • If , where u is a differentiable
    function of x and n is a rational number, then

8
Find the derivative of
  • All of the above

9
DIFFERENTIATING FUNCTIONS INVOLVING RADICALS
  • Example Find the derivative of

10
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11
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12
Evaluate and simplify

13
TRIGONOMETRIC FUNCTIONS AND THE CHAIN RULE
14
DIFFERENTIATING TRIGONOMETRIC FUNCTIONS USING THE
CHAIN RULE
  • Example Find the derivative of

15
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16
DIFFERENTIATING TRIGONOMETRIC FUNCTIONS USING THE
CHAIN RULE
  • Example Find the derivative of

17
(No Transcript)
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