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Unit 3

3.0 Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions

2 min readโ€ขjune 14, 2021

jillian-holbrook

Jillian Holbrook


AP Calculus AB/BCย โ™พ๏ธ

Bookmarkedย 7.1kย โ€ขย 259ย resources
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Exam Weighting โš–

Calculus AB: 9โ€“13%ย 
Calculus BC: 4โ€“7%

Developing Understanding ๐Ÿง 

Unit 3 builds upon understanding from Unit 1 and Unit 2 of AP Calculus. The two main components of Calculus are differentiating and integrating. Of these two, differentiation is the focal point of the first few units in Calculus.ย ย 
This unit, in particular, emphasizes the analysis of functions, continued correct application and understanding of function notation, and the recognition of โ€œinnerโ€ and โ€œouterโ€ functions within composites.ย  This unit makes up 9-13% of the Calculus AB exam and 4-7% of the Calculus BC exam.
This list will be a good place to start in terms of self assessing what you need to study or learn.ย  As you are looking through this list write down what topics on this list you donโ€™t remember or still need to learn.ย  Go to the full study guide page then for this topic to get some more information!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-6Naerh9wxHjq.webp?alt=media&token=6fd3c7c9-9de7-4d83-b593-39c3dd8764d3

Image Courtesy of Giphy

Main Topics and Sections:

Chain Rule (3.1)

Use when you are deriving a function that is a composition of functions:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.49-dB7VLg0PRK1s.png?alt=media&token=ec2db6f0-90be-4964-a10a-00188a2528e0

Resources:

Implicit Differentiation (3.2)

Use when you have an equation that looks like this:
3xy + 2y = 50x
*The first step would be to differentiate with respect to x*

Resources:

Differentiating Inverse Functions (3.3)

Use this formula to help find the derivative of the inverse function:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.50-oHgljxLI27Hx.png?alt=media&token=96ce8297-1eea-4d9b-b5c3-b2cb83d45884

Resources:

Differentiating Inverse Trig Functions (3.4)

Memorize these to help you solve problems quickly and efficiently!
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.52-QfZJ7UxtJOpt.png?alt=media&token=d9e5c7d6-17e7-4488-b46a-e0ef1e4ec48c

Resources:

Selecting Procedures for Calculating Derivatives (3.5)

When you are selecting procedures for calculating derivatives decide if the function you are looking at is:
  • A composition of functions
    • for example:
      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.55-S2786sFs60Ls.png?alt=media&token=4119f594-3450-4853-b68e-66d1a566d791
  • Two functions multiplied together
    • for example:
      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.56-1qtHjFQ2eTSl.png?alt=media&token=688cd1e5-1c41-4005-9f66-d6fcaaeddd6f
  • Two functions divided
    • for example:
      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.57-Nug4LRPaUOhw.png?alt=media&token=171c420a-db39-4dbb-a19b-7e6a4a0af25f
  • A function with exponents
    • for example:
      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.57-mJIEhbXFp4zB.png?alt=media&token=b01c6b93-ebbd-45f2-b8ac-61104d4483ff
  • A function with trig in it
      for example:
      https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202021-06-14%20at%202.58-iGIZk2lYBQaW.png?alt=media&token=686bcfab-862a-42b0-9914-26e14fc90658

Resources:

Calculating Higher Order Derivatives (3.6)

When you get into the next few units you will need to take the next derivate; especially when you get into describing functions and finding where it is increasing, decreasing, concave up and concave down.
The notation for the derivatives are:
  • fโ€™(x): First derivative
  • fโ€™โ€™(x): Second derivative
  • fโ€™โ€™โ€™(x): Third derivativeย 

Resources:

Resources:

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