AP Calculus AB Unit 2: Differentiation: Definition and Basic Rules

Average and instantaneous rates of change, the limit definition of the derivative, estimating a derivative from a table, how differentiability relates to continuity, and the power, sum, product and quotient rules with the derivatives of trig, exponential and logarithmic functions.

CED topics in this unit: 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 2.10.

Key formulas and facts

• $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ and $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$
• Average rate of change on $[a,b]$: $\frac{f(b)-f(a)}{b-a}$
• $\frac{d}{dx}x^n=nx^{n-1}$
• $(fg)'=f'g+fg'$ and $\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}$
• $\frac{d}{dx}\sin x=\cos x$, $\frac{d}{dx}\cos x=-\sin x$, $\frac{d}{dx}\tan x=\sec^2x$, $\frac{d}{dx}\cot x=-\csc^2x$, $\frac{d}{dx}\sec x=\sec x\tan x$, $\frac{d}{dx}\csc x=-\csc x\cot x$
• $\frac{d}{dx}e^x=e^x$ and $\frac{d}{dx}\ln x=\frac1x$
• Differentiable at $c$ $\Rightarrow$ continuous at $c$ (the converse is false)

Worked examples

• Spot a derivative: $\lim_{h\to0}\frac{(2+h)^5-32}{h}$ is $f'(2)$ for $f(x)=x^5$, so it equals $5\cdot2^4=80$
• From a table: if $g(1.8)=4.1$ and $g(2.2)=5.3$, then $g'(2)\approx\frac{5.3-4.1}{0.4}=3$
• Product rule: $\frac{d}{dx}\left(x^3\ln x\right)=3x^2\ln x+x^2$
• Quotient rule: $\frac{d}{dx}\frac{e^x}{x^2+1}=\frac{e^x(x^2+1)-2xe^x}{(x^2+1)^2}=\frac{e^x(x-1)^2}{(x^2+1)^2}$
• Tangent line: $y=4\sqrt x$ at $x=9$ has slope $\frac{2}{\sqrt9}=\frac23$, so $y-12=\frac23(x-9)$

Common traps

• Writing $(fg)'=f'g'$ or $\left(\frac fg\right)'=\frac{f'}{g'}$.
• Rewriting $\frac{3}{x^2}$ as $3x^2$: it is $3x^{-2}$, with derivative $-6x^{-3}$.
• Assuming a continuous function is differentiable: $|x-1|$ has a corner at $x=1$.
• Mixing up the average rate of change (slope between two points) with the instantaneous rate (the derivative at one point).
• Signs on the co-functions: the derivatives of $\cos$, $\cot$ and $\csc$ all start with a minus sign.

Practice questions

69 multiple-choice and 3 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{\cos\left(\frac{\pi}{3}+h\right)-\frac12}{h}$?

1. $\frac{\sqrt{3}}{2}$
2. $- \frac{1}{2}$
3. $- \frac{\sqrt{3}}{2}$
4. $0$

Question 2 · Multiple choice

Let $f(x)=x^3-2x$. Over $1\le x\le3$, how fast does $f$ change on average per unit of $x$?

1. $11$
2. $22$
3. $25$
4. $13$

Question 3 · Multiple choice

The volume $V(t)$, in liters, of water in a container at selected times $t$, in minutes, is shown in the table. Using the data in the table, what is the best estimate of $V'(4)$?

1. $2$
2. $4$
3. $6$
4. $1$

Question 4 · Multiple choice

Which of these functions is continuous at $x=2$ but not differentiable at $x=2$?

1. $|x^3-8|$
2. $\dfrac{x^2-x-2}{x-2}$
3. $(x-2)\,|x-2|$
4. $e^{x-2}$

Question 5 · Multiple choice

If $f(x)=x^2e^x$, what is $f'(1)$?

1. $2 e$
2. $3 e$
3. $e$
4. $3$

Question 6 · Multiple choice

If $y=\dfrac{x}{x^2+4}$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $- \frac{3}{25}$
2. $\frac{1}{2}$
3. $\frac{3}{25}$
4. $\frac{1}{5}$

Question 7 · Multiple choice

If $y=\sec x+\tan x$, what is $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{4}$?

1. $\sqrt{2}$
2. $2$
3. $1 + \sqrt{2}$
4. $\sqrt{2} + 2$

Question 8 · Multiple choice

The line tangent to the graph of $y=\dfrac{8}{\sqrt{x}}$ at $x=4$ crosses the $y$-axis at which value of $y$?

1. $2$
2. $6$
3. $12$
4. $8$

Question 9 · Free response (9 points)

Let $g(x)=\begin{cases}ax^2+1, & x\le 2\\ bx-3, & x>2\end{cases}$ where $a$ and $b$ are constants.

1. (a) Write an equation in $a$ and $b$ that must hold if $g$ is continuous at $x=2$. [1 point]
2. (b) Find the values of $a$ and $b$ for which $g$ is differentiable at $x=2$. [3 points]
3. (c) Using the values from part (b), use the limit definition of the derivative to find $g'(1)$. [3 points]
4. (d) Using the values from part (b), find the average rate of change of $g$ on $[0,4]$, and find a value $c$ in $(0,4)$ where $g'(c)$ equals it. [2 points]

Question 10 · Free response (9 points)

The functions $f$ and $g$ are differentiable for all real numbers. The table gives values of the functions and their derivatives at $x=1$ and $x=2$.

1. (a) Let $h(x)=f(x)\,g(x)$. Find $h'(1)$. [2 points]
2. (b) Let $k(x)=\dfrac{f(x)}{g(x)}$. Find $k'(2)$. [3 points]
3. (c) Let $m(x)=x^2f(x)-3g(x)$. Find $m'(2)$. [2 points]
4. (d) Write an equation of the line tangent to the graph of $g$ at $x=1$, and use it to approximate $g(1.1)$. [2 points]

Question 11 · Multiple choice

Let $f(x)=x^3-x$. What is the slope of the secant line through the points of the graph of $f$ where $x=1$ and $x=3$?

1. $13$
2. $24$
3. $11$
4. $12$

Question 12 · Multiple choice

Let $f(x)=3x^2$. Which of the following is equal to the slope of the line tangent to the graph of $f$ at $x=2$?

1. $\dfrac{3(2.1)^2-12}{0.1}$
2. $\displaystyle\lim_{h\to0}\left(3(2+h)^2-12\right)$
3. $\displaystyle\lim_{h\to0}\frac{3(2+h)^2-12}{2+h}$
4. $\displaystyle\lim_{h\to0}\frac{3(2+h)^2-12}{h}$

Question 13 · Multiple choice

The position of a particle moving along a line is given at selected times in the table. What is the average velocity of the particle over $1\le t\le4$?

1. $\frac{20}{3}$
2. $15$
3. $5$
4. $\frac{11}{2}$

Question 14 · Multiple choice

For $f(x)=x^2+2x$ and $h\neq0$, which expression is equal to $\dfrac{f(-1+h)-f(-1)}{h}$?

1. $2 h$
2. $h + 2$
3. $h - 2$
4. $h$

Question 15 · Multiple choice

Let $g(x)=\ln(x^2+1)$. What is the average rate of change of $g$ over the interval $[0,2]$?

1. $0.805$
2. $1.609$
3. $1.000$
4. $0.402$

Question 16 · Multiple choice

The volume of water in a tank is $V(t)=2t^2+t$ gallons at time $t$ minutes. For $h\neq0$, the average rate of change of $V$ over $[3,\,3+h]$ simplifies to $13+2h$. At what rate is the volume changing at $t=3$?

1. $13$
2. $21$
3. $15$
4. $12$

Question 17 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{(2+h)^5-32}{h}$?

1. $80$
2. $32$
3. $0$
4. $5$

Question 18 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{\cos(\pi+h)+1}{h}$?

1. $-1$
2. $0$
3. $1$
4. The limit does not exist.

Question 19 · Multiple choice

The limit $\displaystyle\lim_{x\to3}\frac{x^2-9}{x-3}$ is the derivative of a function $f$ at $x=a$. Which of the following are $f$ and $a$?

1. $f(x)=x^2$ and $a=9$
2. $f(x)=x^2$ and $a=3$
3. $f(x)=2x$ and $a=3$
4. $f(x)=x^3$ and $a=3$

Question 20 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{e^{1+h}-e}{h}$?

1. $1$
2. $e$
3. $0$
4. The limit does not exist.

Question 21 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{\ln(4+h)-\ln4}{h}$?

1. $4$
2. $\frac{1}{4}$
3. $\ln{\left(4 \right)}$
4. $0$

Question 22 · Multiple choice

The function $f$ is differentiable, and selected values are given in the table. Using the data on the smallest interval centered at $x=2$, what is the best estimate of $f'(2)$?

1. $\frac{18}{5}$
2. $\frac{24}{5}$
3. $7$
4. $\frac{21}{5}$

Question 23 · Multiple choice

The graph of $f$, made of two line segments, is shown. What is $f'(1)$?

1. $-2$
2. $1$
3. $3$
4. $2$

Question 24 · Multiple choice

The graph of $f$, made of two line segments, is shown. What is $f'(-1)+f(-1)$?

1. $3$
2. $-3$
3. $1$
4. $-1$

Question 25 · Multiple choice

A car's velocity $v(t)$, in feet per second, is recorded at selected times $t$ in seconds. Using the data, what is the best estimate of the car's acceleration at $t=4$?

1. $\frac{5}{2}$
2. $3$
3. $10$
4. $1$

Question 26 · Multiple choice

Let $f(x)=e^{\sin x}$. What is the value of $f'(1)$?

1. $2.320$
2. $1.253$
3. $0.540$
4. $1.000$

Question 27 · Multiple choice

The graph of $f$ and the line tangent to the graph at $x=2$ (dashed) are shown. The tangent line passes through $(0,2)$ and $(6,5)$. What is $f'(2)$?

1. $3$
2. $\frac{1}{2}$
3. $2$
4. $\frac{3}{2}$

Question 28 · Multiple choice

Let $f(x)=|x-3|$. At $x=3$, $f$ is

1. continuous but not differentiable
2. continuous and differentiable
3. differentiable but not continuous
4. neither continuous nor differentiable

Question 29 · Multiple choice

The function $f$ is defined by $f(x)=x^2+1$ for $x\le1$ and $f(x)=2x$ for $x>1$. At $x=1$, $f$ is

1. continuous but not differentiable
2. differentiable but not continuous
3. neither continuous nor differentiable
4. continuous and differentiable

Question 30 · Multiple choice

Let $f(x)=\begin{cases}x^2, & x\le1\\ 3x-2, & x>1.\end{cases}$ At $x=1$, $f$ is

1. continuous and differentiable
2. differentiable but not continuous
3. neither continuous nor differentiable
4. continuous but not differentiable

Question 31 · Multiple choice

Let $f(x)=\sqrt[3]{x}$. At $x=0$, $f$ is

1. continuous and differentiable
2. continuous but not differentiable
3. differentiable but not continuous
4. neither continuous nor differentiable

Question 32 · Multiple choice

The graph of $f$, made of line segments, is shown. At how many values of $x$ in the open interval $(-2,4)$ is $f$ not differentiable?

1. $1$
2. $3$
3. $2$
4. $0$

Question 33 · Multiple choice

Let $f(x)=\begin{cases}ax^2+b, & x<2\\ 4x, & x\ge2.\end{cases}$ If $f$ is differentiable at $x=2$, what is the value of $b$?

1. $1$
2. $8$
3. $4$
4. $0$

Question 34 · Multiple choice

If $f(x)=6x^{2/3}$, what is $f'(8)$?

1. $4$
2. $8$
3. $1$
4. $2$

Question 35 · Multiple choice

If $y=\dfrac{1}{\sqrt{x}}$, what is $\dfrac{dy}{dx}$ at $x=4$?

1. $- \frac{1}{4}$
2. $\frac{1}{16}$
3. $- \frac{1}{8}$
4. $- \frac{1}{16}$

Question 36 · Multiple choice

If $f(x)=\dfrac{1}{5x^2}$, which of the following is $f'(x)$?

1. $- \frac{2}{5 x^{3}}$
2. $- \frac{1}{10 x}$
3. $\frac{2}{5 x^{3}}$
4. $- \frac{2}{5 x}$

Question 37 · Multiple choice

What is the slope of the line tangent to the graph of $y=x^{3/2}$ at $x=9$?

1. $27$
2. $\frac{3}{2}$
3. $\frac{9}{2}$
4. $9$

Question 38 · Multiple choice

At what value of $x$ does the graph of $y=x^4$ have a tangent line with slope $32$?

1. $8$
2. $2$
3. $16$
4. $4$

Question 39 · Multiple choice

Which of the following is an equation of the line tangent to the graph of $y=x^3$ at $x=2$?

1. $12 x - 24$
2. $3 x + 2$
3. $12 x - 16$
4. $12 x + 8$

Question 40 · Multiple choice

If $f(x)=4x^3-5x+7$, what is $f'(-1)$?

1. $7$
2. $17$
3. $-1$
4. $8$

Question 41 · Multiple choice

If $y=3\sqrt{x}-\dfrac{2}{x}$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $- \frac{1}{2}$
2. $\frac{5}{2}$
3. $1$
4. $\frac{7}{2}$

Question 42 · Multiple choice

If $g(x)=(x^2+1)^2$, what is $g'(1)$?

1. $4$
2. $2$
3. $8$
4. $16$

Question 43 · Multiple choice

If $f(x)=\dfrac{x^3-4x}{x}$ for $x\neq0$, what is $f'(3)$?

1. $6$
2. $23$
3. $5$
4. $27$

Question 44 · Multiple choice

Functions $f$ and $g$ are differentiable with $f'(2)=5$ and $g'(2)=-1$. If $h(x)=3f(x)-2g(x)+x^2$, what is $h'(2)$?

1. $17$
2. $13$
3. $21$
4. $19$

Question 45 · Multiple choice

At which values of $x$ does the graph of $y=x^3-12x+1$ have a horizontal tangent line?

1. $x=2$ only
2. $x=0$ only
3. $x=-2\sqrt3$ and $x=2\sqrt3$
4. $x=-2$ and $x=2$

Question 46 · Multiple choice

If $f(x)=2\sin x-3\cos x$, what is $f'\!\left(\frac{\pi}{2}\right)$?

1. $-3$
2. $2$
3. $-2$
4. $3$

Question 47 · Multiple choice

For $f(x)=2e^x-\ln x$, the value of $f'(1)$ is

1. $2 e$
2. $1 + 2 e$
3. $-1 + 2 e$
4. $1$

Question 48 · Multiple choice

What is the slope of the line tangent to $y=5\ln x+x$ at $x=5$?

1. $6$
2. $2$
3. $1$
4. $1 + 5 \ln{\left(5 \right)}$

Question 49 · Multiple choice

Which of the following is an equation of the line tangent to the graph of $y=e^x$ at $x=0$?

1. $x$
2. $e x + 1$
3. $x + 1$
4. $1$

Question 50 · Multiple choice

Let $f(x)=\sin x+\cos x$. For what value of $x$ in the open interval $(0,\pi)$ is $f'(x)=0$?

1. $\frac{\pi}{4}$
2. $\frac{\pi}{2}$
3. $\frac{3 \pi}{4}$
4. $\frac{\pi}{3}$

Question 51 · Multiple choice

What is $\displaystyle\lim_{h\to0}\frac{\sin\left(\frac{\pi}{6}+h\right)-\frac12}{h}$?

1. $\frac{1}{2}$
2. $\frac{\sqrt{3}}{2}$
3. $- \frac{\sqrt{3}}{2}$
4. $0$

Question 52 · Multiple choice

If $f(x)=3e^x-2\ln x$, what is $f'(2)$?

1. $-2 + 3 e^{2}$
2. $3 e^{2}$
3. $1 + 3 e^{2}$
4. $-1 + 3 e^{2}$

Question 53 · Multiple choice

If $f(x)=x^2\sin x$, what is $f'(\pi)$?

1. $\pi^{2}$
2. $- \pi^{2}$
3. $2 \pi$
4. $0$

Question 54 · Multiple choice

The table gives values of differentiable functions $f$ and $g$ and their derivatives at $x=3$. If $h(x)=f(x)\,g(x)$, what is $h'(3)$?

1. $-5$
2. $6$
3. $14$
4. $-4$

Question 55 · Multiple choice

At what value of $x$ does the graph of $y=xe^x$ have a horizontal tangent line?

1. $0$
2. $-1$
3. $1$
4. $- e$

Question 56 · Multiple choice

If $f(x)=\sqrt{x}\,\ln x$, what is $f'(e^2)$?

1. $e^{-1}$
2. $\frac{2}{e}$
3. $\frac{1}{2}$
4. $\frac{3}{e}$

Question 57 · Multiple choice

If $y=(2x+1)(x^2-3)$, what is $\dfrac{dy}{dx}$ at $x=1$?

1. $12$
2. $-4$
3. $2$
4. $-6$

Question 58 · Multiple choice

If $f(x)=e^x\cos x$, what is $f'(0)$?

1. $0$
2. $-1$
3. $2$
4. $1$

Question 59 · Multiple choice

If $f(x)=\dfrac{x+1}{x-1}$, what is $f'(3)$?

1. $\frac{1}{2}$
2. $- \frac{1}{2}$
3. $- \frac{1}{4}$
4. $2$

Question 60 · Multiple choice

If $g(x)=\dfrac{x^2}{e^x}$, what is $g'(1)$?

1. $\frac{3}{e}$
2. $2$
3. $- \frac{1}{e}$
4. $e^{-1}$

Question 61 · Multiple choice

The table gives values of differentiable functions $f$ and $g$ and their derivatives at $x=2$. What is the value of $\dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)$ at $x=2$?

1. $11$
2. $13$
3. $-13$
4. $\frac{1}{4}$

Question 62 · Multiple choice

At what value of $x$ does the graph of $y=\dfrac{\ln x}{x}$ have a horizontal tangent line?

1. $1$
2. $e^{-1}$
3. $e^{2}$
4. $e$

Question 63 · Multiple choice

If $f(x)=\dfrac{3x-2}{x^2+1}$, what is $f'(0)$?

1. $3$
2. $-2$
3. $0$
4. $-3$

Question 64 · Multiple choice

If $y=\dfrac{\sin x}{1+\cos x}$, what is $\dfrac{dy}{dx}$ at $x=\frac{\pi}{2}$?

1. $1$
2. $0$
3. $-1$
4. $\frac{1}{2}$

Question 65 · Multiple choice

If $f(x)=\tan x$, what is $f'\!\left(\frac{\pi}{3}\right)$?

1. $4$
2. $\sqrt{3}$
3. $2$
4. $\frac{4}{3}$

Question 66 · Multiple choice

If $y=\sec x$, what is $\dfrac{dy}{dx}$ at $x=\frac{\pi}{4}$?

1. $2$
2. $1$
3. $\frac{\sqrt{2}}{2}$
4. $\sqrt{2}$

Question 67 · Multiple choice

If $f(x)=\csc x$, what is $f'\!\left(\frac{\pi}{6}\right)$?

1. $2 \sqrt{3}$
2. $- 2 \sqrt{3}$
3. $-4$
4. $- \frac{\sqrt{3}}{2}$

Question 68 · Multiple choice

If $g(x)=\cot x$, what is $g'\!\left(\frac{\pi}{4}\right)$?

1. $2$
2. $-1$
3. $- \sqrt{2}$
4. $-2$

Question 69 · Multiple choice

Which of the following is an equation of the line tangent to the graph of $y=\tan x$ at $x=\frac{\pi}{4}$?

1. $2 x - \frac{\pi}{2} + 1$
2. $x - \frac{\pi}{4} + 1$
3. $2 x - \frac{\pi}{2}$
4. $2 x + 1$

Question 70 · Multiple choice

If $f(x)=x\tan x$, what is $f'\!\left(\frac{\pi}{4}\right)$?

1. $\frac{\pi}{2}$
2. $1 + \frac{\pi}{2}$
3. $\frac{\pi}{4} + 1$
4. $2$

Question 71 · Multiple choice

If $h(x)=3\tan x-2\sec x$, what is $h'(0)$?

1. $1$
2. $-2$
3. $5$
4. $3$

Question 72 · Free response (9 points)

Let $g$ be the function defined by $g(x)=\begin{cases}x^2-2x+3, & x\le2\\ ax+b, & x>2,\end{cases}$ where $a$ and $b$ are constants, and $g$ is differentiable at $x=2$.

1. (a) Find the values of $a$ and $b$. Show the work that leads to your answer. [3 points]
2. (b) Use the definition of the derivative to find $g'(1)$. [2 points]
3. (c) Find the average rate of change of $g$ on the interval $[0,3]$. [2 points]
4. (d) Write an equation for the line tangent to the graph of $g$ at $x=3$. [2 points]