Free AP Calc AB Unit 3 Practice Test: Chain Rule, Implicit and Inverse Derivatives (PDF)

This free AP Calculus AB Unit 3 practice test covers Differentiation: Composite, Implicit, and Inverse Functions only: the chain rule from formulas, tables and graphs, implicit differentiation and tangent lines, derivatives of inverse functions, the derivatives of arcsin, arccos and arctan, choosing the right procedure, and second derivatives, including implicit ones. It has 16 four-choice multiple-choice questions and one 9-point free-response question on an implicitly defined curve, 25 points in 45 minutes with no calculator. Every explanation shows the working and says why each wrong answer is wrong.

1. Question 11 mark · Chain rule
   The table gives selected values of the differentiable functions $f$ and $g$ and their derivatives. If $h(x) = f(g(x))$, what is the value of $h^{\prime}(2)$?
   $$\begin{array}{c|c|c|c|c} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 1 & 6 & -4 & 0 & 3 \\ 2 & 3 & -3 & 4 & 5 \\ 3 & 1 & -2 & 7 & 2 \\ 4 & -1 & -2 & 8 & 0 \\ \hline \end{array}$$
   • A.
     $-15$
   • B.
     $-2$
   • C.
     $-10$
   • D.
     $-6$
2. Question 21 mark · Chain rule
   The graph of the function $g$ on the interval $0 \leq x \leq 6$ consists of two line segments: one from $(0, 1)$ to $(3, 7)$ and one from $(3, 7)$ to $(6, 4)$. If $h(x) = g(x^2)$, what is the value of $h^{\prime}(2)$?
   • A.
     $-4$
   • B.
     $-1$
   • C.
     $8$
   • D.
     $24$
3. Question 31 mark · Chain rule
   If $y = \cos^3(2x)$, then $\dfrac{dy}{dx} =$
   • A.
     $3\cos^2(2x)$
   • B.
     $-3\cos^2(2x)\sin(2x)$
   • C.
     $6\cos^2(2x)\sin(2x)$
   • D.
     $-6\cos^2(2x)\sin(2x)$
4. Question 41 mark · Chain rule
   If $f(x) = \sqrt{2x^2 + 1}$, what is the slope of the line tangent to the graph of $f$ at $x = 2$?
   • A.
     $\frac{1}{6}$
   • B.
     $\frac{4}{3}$
   • C.
     $\frac{8}{3}$
   • D.
     $\frac{2}{3}$
5. Question 51 mark · Implicit differentiation
   The curve defined by $x^2 y + y^3 = 10$ passes through the point $(3, 1)$. What is the value of $\dfrac{dy}{dx}$ at $(3, 1)$?
   • A.
     $-2$
   • B.
     $-\frac{1}{2}$
   • C.
     $-1$
   • D.
     $0$
6. Question 61 mark · Implicit differentiation
   At which of the following points on the curve $x^2 + 2xy + 3y^2 = 6$ is the line tangent to the curve horizontal?
   • A.
     $(0, \sqrt{2})$
   • B.
     $(3, -1)$
   • C.
     $(1, -1)$
   • D.
     $(\sqrt{3}, -\sqrt{3})$
7. Question 71 mark · Implicit differentiation
   Which of the following is an equation of the line tangent to the curve $x e^{y} + y = 2$ at the point $(2, 0)$?
   • A.
     $y = -\frac{1}{3}(x - 2)$
   • B.
     $y = -\frac{1}{2}(x - 2)$
   • C.
     $y = -3(x - 2)$
   • D.
     $y = 3(x - 2)$
8. Question 81 mark · Derivatives of inverse functions
   Let $f(x) = x^3 + 2x + 1$, and let $g$ be the inverse function of $f$. What is the value of $g^{\prime}(4)$?
   • A.
     $\frac{1}{50}$
   • B.
     $5$
   • C.
     $\frac{1}{5}$
   • D.
     $1$
9. Question 91 mark · Derivatives of inverse functions
   The function $f$ is differentiable and increasing, and selected values of $f$ and its derivative $f^{\prime}$ are given in the table. If $g$ is the inverse function of $f$, what is the value of $g^{\prime}(5)$?
   $$\begin{array}{c|c|c|c|c} \hline x & 1 & 2 & 3 & 5 \\ \hline f(x) & 0 & 2 & 5 & 15 \\ \hline f^{\prime}(x) & 1 & 3 & 4 & 6 \\ \hline \end{array}$$
   • A.
     $\frac{1}{4}$
   • B.
     $\frac{1}{6}$
   • C.
     $3$
   • D.
     $4$
10. Question 101 mark · Derivatives of inverse functions
    The function $f$ is differentiable and one-to-one, with $f(2) = 7$ and $f^{\prime}(2) = -3$. Which of the following is an equation of the line tangent to the graph of $f^{-1}$ at $x = 7$?
    • A.
      $y - 7 = -\frac{1}{3}(x - 2)$
    • B.
      $y - 2 = -3(x - 7)$
    • C.
      $y - 2 = \frac{1}{3}(x - 7)$
    • D.
      $y - 2 = -\frac{1}{3}(x - 7)$
11. Question 111 mark · Inverse trig derivatives
    $\dfrac{d}{dx}\left[\arctan(3x)\right] =$
    • A.
      $\dfrac{1}{1 + 9x^2}$
    • B.
      $\dfrac{3}{1 + 9x^2}$
    • C.
      $\dfrac{3}{1 + 3x^2}$
    • D.
      $3\sec^2(3x)$
12. Question 121 mark · Inverse trig derivatives
    If $f(x) = \arccos(2x)$, then $f^{\prime}\left(\frac{1}{4}\right) =$
    • A.
      $\frac{4\sqrt{3}}{3}$
    • B.
      $-\frac{2\sqrt{3}}{3}$
    • C.
      $-\frac{4\sqrt{3}}{3}$
    • D.
      $-\frac{8\sqrt{15}}{15}$
13. Question 131 mark · Selecting derivative procedures
    If $f(x) = \ln\left(\dfrac{x^2}{x + 1}\right)$ for $x > 0$, then $f^{\prime}(1) =$
    • A.
      $2$
    • B.
      $\frac{3}{4}$
    • C.
      $\frac{5}{2}$
    • D.
      $\frac{3}{2}$
14. Question 141 mark · Selecting derivative procedures
    If $h(x) = x \arcsin x$, then $h^{\prime}\left(\frac{1}{2}\right) =$
    • A.
      $\frac{\pi}{6} + \frac{\sqrt{3}}{3}$
    • B.
      $\frac{2\sqrt{3}}{3}$
    • C.
      $\frac{\pi}{6} + \frac{2\sqrt{3}}{3}$
    • D.
      $\frac{\pi}{3} + \frac{\sqrt{3}}{3}$
15. Question 151 mark · Higher-order derivatives
    For the curve $x^2 - y^2 = 16$, what is the value of $\dfrac{d^2y}{dx^2}$ at the point $(5, 3)$?
    • A.
      $\frac{1}{3}$
    • B.
      $\frac{16}{27}$
    • C.
      $-\frac{16}{27}$
    • D.
      $-\frac{16}{9}$
16. Question 161 mark · Higher-order derivatives
    If $f(x) = \ln(1 + x^2)$, then $f^{\prime\prime}(2) =$
    • A.
      $\frac{6}{25}$
    • B.
      $-\frac{6}{25}$
    • C.
      $\frac{2}{5}$
    • D.
      $-\frac{6}{5}$
17. Question 179 marks · Implicit differentiation & tangent lines
    Implicit differentiation. Consider the curve given by the equation $x^3 - 2xy + y^2 = 4$. The point $(1, 3)$ lies on the curve.
    $$x^3 - 2xy + y^2 = 4$$
    (a)

    Show that $\dfrac{dy}{dx} = \dfrac{3x^2 - 2y}{2x - 2y}$.

    [2]
    (b)

    Write an equation for the line tangent to the curve at the point $(1, 3)$.

    [2]
    (c)

    Find the coordinates of every point on the curve at which the line tangent to the curve is vertical. Justify your answer.

    [3]
    (d)

    Evaluate $\dfrac{d^2y}{dx^2}$ at the point $(1, 3)$.

    [2]

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