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Differentiation: Composite, Implicit, and Inverse Functions

Six topics on differentiating the functions that are not handed to you in a convenient form. The chain rule and the inner factor everyone drops, implicit differentiation where every y-term carries a dy/dx, the reciprocal slopes of an inverse and the input they are evaluated at, the six inverse trigonometric forms in three mirrored pairs, the discipline of choosing a procedure before applying one, and higher-order derivatives, where every rule has to be applied again.

AB exam 5-10%BC exam 5-10%6 topics
Topics
Key forms For every problem in this unit
Chain rule
d/dx f(g(x)) = f′(g(x)) · g′(x)
Leibniz form
dy/dx = (dy/du) · (du/dx)
Power with an inside
d/dx [un] = n un−1 · u′
Implicit: any y-term
d/dx [y2] = 2y · dy/dx
Implicit: mixed term
d/dx [xy] = y + x · dy/dx
Second derivative
differentiate y′ again, then substitute y′ back in
Inverse function
(f−1)′(x) = 1 / f′(f−1(x))
arcsin x
1 / √(1 − x2)
arccos x
−1 / √(1 − x2)
arctan x
1 / (1 + x2)
arccot x
−1 / (1 + x2)
arcsec x
1 / (|x| √(x2 − 1))
With an inside u
every form above gains a factor of u′ on top
Unit 3 tools
Challenge bank
1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

0 mastered · 0 to revisit · 60 total
Question
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Worked solution
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Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 6 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

20questions
6topics
10codes covered
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Course so far · AB

Check what stuck.

Units 1 through 3, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 45-question section, your choice. Even coverage means this is a retention check rather than a score estimate. Follows the track you chose on the course page.

20 or 45questions
32topics
40codes covered
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