Unit 3: Composite & Implicit Differentiation
Calculus AB · Unit 3 · Paper 2

Composite & Implicit Differentiation unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1Paper 2Paper 3
Each paper is built from this unit’s 38 terms and is the same for everyone, so a teacher can assign “Unit 3, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.
Suggested time 36 min 33 points0/17 attempted
1

Where to evaluate an inverse derivative

2

Higher-order derivatives of sine

3

Implicit second derivative sign

4

Chain rule with a logarithm

5

Related rates as implicit differentiation

6

Vertical tangent from an implicit derivative

7

Chain rule

8

Inverse derivative formula

9

Chain rule with an exponential

10

Derivative of a variable base with a variable exponent

11

Chain rule with a table

12

Vertical and horizontal tangents implicitly

Short answer 1. Define or explain: Why dy/dx involves y

3 pts

Short answer 2. Define or explain: Candidate points must lie on the curve

3 pts

Short answer 3. Define or explain: Sign of an implicit second derivative

3 pts

Short answer 4. Define or explain: Logarithmic differentiation

3 pts

Free response

9 pts

Consider the curve defined implicitly by x² + 2xy − y² = 7.

Show that the point (2, 1) lies on the curve, and find dy/dx in terms of x and y.

Write the equation of the line tangent to the curve at (2, 1).

Find the coordinates of all points on the curve at which the tangent line is vertical, and justify that the tangent is vertical rather than undefined for some other reason.