Unit 2 · Topic 2.7 · about 30 minutes · 7 parts · 12 scored checks, 3 of them multiple choice, plus 1 from earlier lessons

AP Calculus AB 2.7: Derivatives of cos x, sin x, eˣ, and ln x

Q: Why do the sine and cosine rules need radians?

They come from h 0 hh = 1, which holds when h is in radians. Measured in degrees, the same limit is 180, and every slope of sine and cosine picks up that factor.

Q: What is the derivative of x at x = 0 or at negative x?

There is none. x is defined only for x > 0, and a function has no derivative where it has no value. The rule ddx x = 1x is for x > 0.

Q: Does the minus sign go with the derivative of sine or of cosine?

Cosine: ddx x = - x. To check, picture the cosine graph just to the right of 0, where it falls from its highest value. A falling graph has a negative slope.

Built by Tanner Crow from College Board's course framework · Every answer key independently checked

Unit 2: 10 to 15% of the multiple-choice section

What you'll be able to do

Find derivatives of $\sin x$ , $\cos x$ , $e^{x}$ and $\ln x$ , and of sums and constant multiples of them, with the rule for each.
Evaluate a limit by recognizing it as the definition of the derivative of a function whose derivative you know.

Key vocabulary

Term	What it means	The exam trap
Radian measure	Angle measure in which one full turn is $2 π$ , so one radian is $\frac{180}{π}$ degrees, about 57.3 degrees.	$π$ radians is a half turn, 180 degrees, so read $\cos π$ as the cosine of a half turn, not of about 3 degrees.
Natural exponential function, $e^{x}$	The exponential function whose base is the number $e \approx 2.71828$ .	Do not confuse $e^{x}$ , a fixed base with a variable exponent, with $x^{e}$ , a variable base with a fixed exponent.
Natural logarithm, $\ln x$	The logarithm with base $e$ , defined only for $x > 0$ .	$\ln x$ has no value at $x = 0$ or below, so neither does its derivative; check the domain before you evaluate.
Definition of the derivative	$f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$ , or equally $\lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ , when the limit exists.	A limit with this shape may write $f (a)$ as a plain number, so ask whether the number on top is a value of the function.

Graph of y = ln x for x from 0.15 to 6.2, with tangent lines drawn at x = 1, 2 and 4. Their slopes are 1, 0.5 and 0.25, which equal 1 divided by x at each point.

Part 1 Think first How fast is the buoy moving?

Guess before you learn. A wrong guess still makes the answer stick.

Predict first

A buoy rides up and down on ocean swells. Its height above its resting level, in meters, $t$ seconds into the motion is $\sin t$ . Over the short interval from $t$ to $t + 0.001$ , its average rate of change is 1.000 meter per second at $t = 0$ , about 0 at $t = \frac{π}{2}$ , $- 1.000$ at $t = π$ and 0.540 at $t = 1$ . Which function of $t$ gives the buoy's rate of change at each instant?

In short. Sine, cosine, the exponential function $e^{x}$ and the natural logarithm $\ln x$ each have a derivative rule of their own, and a limit shaped like the definition of the derivative can be evaluated with those rules.

The rules for $\sin x$ and $\cos x$ hold only with $x$ in radians, and the derivative of cosine carries a minus sign.
$e^{x}$ is its own derivative, and $\ln x$ , defined for $x > 0$ , has derivative $\frac{1}{x}$ .
A limit shaped like $\lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$ for a function $f$ you can differentiate equals $f^{'} (a)$ .

Part 2 Learn it Three short ideas, each checked as you go

7a. Sine and cosine, in radians

In Think first, the buoy's average rates matched $\cos t$ at every input tried. That pattern holds at every $x$ :

\frac{d}{d x} \sin x = \cos x and \frac{d}{d x} \cos x = - \sin x .

The minus sign belongs to the cosine rule, and you can see why. The graph of $y = \cos x$ is at its highest value, 1, at $x = 0$ , so just to the right of 0 it is falling and its slope is negative. $- \sin x$ is negative for small positive $x$ , so the sign fits.

Both rules assume $x$ is measured in radians. They rest on the limit $\lim_{h \to 0} \frac{\sin h}{h} = 1$ , which is true only in radians. On a calculator in degree mode, the slope of $y = \sin x$ at $x = 0$ comes out near 0.0175, not 1, because one degree is only $\frac{π}{180}$ of a radian. Every angle in this course is in radians, so keep the calculator in radian mode.

The rules from 2.6 still apply: a constant multiple comes along, and a sum is differentiated term by term. For example,

\frac{d}{d x} (3 \cos x + x^{2}) = - 3 \sin x + 2 x .

Quick check · Match

No calculator. Tap a function, then its derivative.

7b. The exponential and the natural logarithm

The number $e \approx 2.71828$ is the base that makes an exponential function its own derivative:

\frac{d}{d x} e^{x} = e^{x} .

At every point of $y = e^{x}$ , the slope equals the height. At $x = 0$ both are 1, and at $x = 2$ both are $e^{2}$ .

Do not reach for the power rule here. It is for $x^{n}$ , a variable raised to a fixed number. In $e^{x}$ the base is fixed and the variable is the exponent, so $x e^{x - 1}$ is not its derivative.

The natural logarithm $\ln x$ is the logarithm with base $e$ . It is defined only for $x > 0$ , and there

\frac{d}{d x} \ln x = \frac{1}{x} .

The graph at the top shows it: the tangent lines to $y = \ln x$ at $x = 1$ , 2 and 4 have slopes 1, 0.5 and 0.25.

With the rules from 2.6, $g (x) = e^{x} + 6 \ln x$ has $g^{'} (x) = e^{x} + \frac{6}{x}$ , so $g^{'} (2) = e^{2} + 3$ .

Differentiate first, then substitute. $\ln 1 = 0$ , but the slope of $y = \ln x$ at $x = 1$ is 1, not 0. A term whose value is 0 at a point can still be changing there.

Quick check · Multiple choice

No calculator. Let $f (x) = 3 e^{x} - 5 \ln x$ . What is $f^{'} (1)$ ?

7c. Limits that are derivatives in disguise

The definition of the derivative from 2.1 says

f^{'} (a) = \lim_{h \to 0} \frac{f (a + h) - f (a)}{h} = \lim_{x \to a} \frac{f (x) - f (a)}{x - a} .

This holds provided the limit exists.

Read it backward and it evaluates limits. Take $\lim_{h \to 0} \frac{\ln (4 + h) - \ln 4}{h}$ . Substituting $h = 0$ gives $\frac{0}{0}$ , which settles nothing. But the expression has exactly the shape of the definition, with $f (x) = \ln x$ and $a = 4$ . So the limit is $f^{'} (4)$ , and the rule gives $f^{'} (4) = \frac{1}{4}$ , with no algebra at all.

Sometimes $f (a)$ is written as a plain number, and you have to recognize it. In

\lim_{x \to π} \frac{\sin x}{x - π},

the top is $\sin x - 0$ , and $0 = \sin π$ . So this is $\lim_{x \to π} \frac{\sin x - \sin π}{x - π}$ , the derivative of $\sin x$ at $π$ , which is $\cos π = - 1$ .

To spot one, look for the same function at two inputs on top, one of them the fixed point $a$ , and the difference of those two inputs underneath. Then the limit is the value of a derivative you already know.

Quick check · Fill the statement

No calculator. Tap a blank, then tap an answer.

The limit $\lim_{h \to 0} \frac{e^{2 + h} - e^{2}}{h}$ is $f^{'} (a)$ for the function $f (x) =$ at the point $a =$ , so the limit equals .

Common mistakes

The mistake	Why it is wrong	Do this instead
Using the power rule on $e^{x}$ and writing its derivative as $x e^{x - 1}$ .	The power rule is for $x^{n}$ , where the exponent is a fixed number. In $e^{x}$ the exponent is the variable.	Use $\frac{d}{d x} e^{x} = e^{x}$ . Keep the power rule for terms like $x^{3}$ or $x^{- 1}$ .
Dropping the minus sign and writing the derivative of $\cos x$ as $\sin x$ .	Cosine falls just to the right of $x = 0$ , so its slope there is negative, and $\sin x$ is positive there.	Write $\frac{d}{d x} \cos x = - \sin x$ , then carry any constant in front with its own sign.
Substituting the number before differentiating, so a term such as $5 \ln x$ at $x = 1$ disappears because $\ln 1 = 0$ .	A term's value at a point says nothing about how fast it is changing there.	Find $f^{'} (x)$ first, then substitute into $f^{'} (x)$ .

Part 3 Watch it done A limit that is a derivative

Watch each step, then do the last one yourself.

Evaluate $\lim_{h \to 0} \frac{\sin (\frac{π}{3} + h) - \frac{\sqrt{3}}{2}}{h}$ .

Step 1 · Name the number on top
$\frac{\sqrt{3}}{2} = \sin \frac{π}{3}$ , so the top is $\sin (\frac{π}{3} + h) - \sin \frac{π}{3}$ .
Why this step?
Why rewrite $\frac{\sqrt{3}}{2}$ as $\sin \frac{π}{3}$ ?
The definition needs the same function at two inputs on top, $f (a + h) - f (a)$ . As a bare number, $\frac{\sqrt{3}}{2}$ hides that the second term is the sine at $\frac{π}{3}$ , the same point that appears in the first term.
Step 2 · Match the definition
The limit is $\lim_{h \to 0} \frac{f (a + h) - f (a)}{h}$ with $f (x) = \sin x$ and $a = \frac{π}{3}$ , so it equals $f^{'} (\frac{π}{3})$ .
Step 3 · Differentiate by the rule
$f^{'} (x) = \cos x$ , so the limit is $\cos \frac{π}{3}$ .
Your step · Calculate

No calculator. Finish it. The limit $\lim_{h \to 0} \frac{\sin (\frac{π}{3} + h) - \frac{\sqrt{3}}{2}}{h}$ equals $\cos \frac{π}{3}$ . What is its value? Give an exact decimal.

Your answer

Part 4 Try it New functions, both ideas

Try it · Sort it

No calculator. Tap a function, then the sign of its derivative at $x = 0$ .

Positive

Zero

Negative

Try it · Multiple choice

No calculator. What is $\lim_{h \to 0} \frac{\cos (\frac{π}{2} + h)}{h}$ ?

Part 5 Explain it Write it the way the exam asks

On the exam free response is handwritten, so work on paper first, with your graphing calculator in radian mode. Nothing you type here leaves this page. Then compare with the model answer and score each point.

Free response is 50% of the exam score. This one is shaped like Question 1, from free-response Part A (graphing calculator required).

Free-response practice · Question 1 style

Graphing calculator required. The depth of the water at the end of a dock, in feet, is modeled by $D (t) = 9 + 2 \sin t + \ln t$ , where $t$ is measured in hours after midnight and $1 \leq t \leq 7$ . Your calculator is in radian mode. Give each decimal answer to three decimal places.

(a) Find $D^{'} (t)$ .

(b) Find the value of $\lim_{h \to 0} \frac{D (5 + h) - D (5)}{h}$ . Using correct units, interpret the meaning of this value in the context of the problem.

(c) Find all times $t$ , for $1 \leq t \leq 7$ , at which the depth of the water is increasing at a rate of 1 foot per hour. Show the equation you solve.

Your answer stays on this page. It is never sent or saved.

Model answer

(a) $D^{'} (t) = 0 + 2 \cos t + \frac{1}{t} = 2 \cos t + \frac{1}{t}$ .

(b) The limit is the definition of the derivative of $D$ at $t = 5$ , so it equals $D^{'} (5) = 2 \cos 5 + \frac{1}{5} = 0.767$ . At $t = 5$ , 5 a.m., the depth of the water is increasing at a rate of 0.767 feet per hour.

(c) The depth is increasing at 1 foot per hour where $D^{'} (t) = 1$ , so solve $2 \cos t + \frac{1}{t} = 1$ on the calculator for $1 \leq t \leq 7$ . The solutions are $t = 1.422$ and $t = 5.127$ hours after midnight (truncated, 1.421 and 5.126).

Score your own answer. This is a self-check, not a grade. Tick each point your answer makes.

Writes $D^{'} (t) = 2 \cos t + \frac{1}{t}$ , with the derivative of the constant 9 as 0.
Names the limit in (b) as $D^{'} (5)$ and gets 0.767.
Says the depth is increasing at 0.767 feet per hour at $t = 5$ , with units.
Writes the equation $2 \cos t + \frac{1}{t} = 1$ before using the calculator.
Gets both times, $t = 1.422$ and $t = 5.127$ (or 1.421 and 5.126 truncated).

Part 6 Check yourself A ride on a Ferris wheel

The three case questions use the same model and table. Answer them from it.

A car on a Ferris wheel passes its lowest point at $t = 0$ . Its height above the ground, in meters, $t$ minutes later is $H (t) = 12 - 10 \cos t$ . Some values are in the table.

Height of the car above the ground
$t$ (minutes)	$H (t)$ (meters)
$0$	2
$\frac{π}{2}$	12
$π$	22
$\frac{3 π}{2}$	12

Case question 1 · Calculate

No calculator. Find $H^{'} (\frac{π}{6})$ , the rate at which the car's height is changing at $t = \frac{π}{6}$ , in meters per minute.

Your answer

Case question 2 · Multiple choice

No calculator. Which statement describes the car at $t = \frac{3 π}{2}$ minutes?

Case question 3 · Calculate

No calculator. Use the table. What is $\lim_{t \to π / 2} \frac{H (t) - 12}{t - \frac{π}{2}}$ ?

Your answer

The rest stand on their own and do not use the case.

Check yourself · Spot the error

No calculator. A student finds $f^{'} (\frac{π}{2})$ for $f (x) = 4 \cos x - 3 \ln x + e^{x}$ . Tap the first line with a mistake.

Check yourself · Decide and justify

No calculator. A classmate's calculator is in degree mode. It estimates the derivative of $y = \sin x$ at $x = 0$ as 0.017453. The classmate says the rule $\frac{d}{d x} \sin x = \cos x$ must be wrong, because $\cos 0 = 1$ . Is the classmate right?

Which reason supports your answer?

Part 7 Keep it Bring back earlier ideas

Every rule in this lesson combines with the rules of 2.6, and the sine and cosine rules rest on a limit that the squeeze theorem of Unit 1 settles. Pulling them back up is what makes them last.

Bring it back · Calculate

No calculator. Let $g (x) = 4 x^{3} - 7 x^{2} + 2 x - 5$ . Find $g^{'} (- 1)$ .

Your answer

From 1.8 Determining Limits Using the Squeeze Theorem · Calculate

No calculator. For every $x \neq 3$ , the function $f$ satisfies $6 x - 9 \leq f (x) \leq x^{2}$ . Find $\lim_{x \to 3} f (x)$ .

Your answer

In three lines

With $x$ in radians, $\frac{d}{d x} \sin x = \cos x$ and $\frac{d}{d x} \cos x = - \sin x$ .
$\frac{d}{d x} e^{x} = e^{x}$ , and $\frac{d}{d x} \ln x = \frac{1}{x}$ for $x > 0$ ; differentiate before you substitute.
A limit shaped like the definition of $f^{'} (a)$ for a known $f$ is evaluated by finding $f^{'} (a)$ with a rule.

Exam strategy for this topic

Section I Part A, Multiple Choice (calculator not permitted): 29 questions in 62 minutes.
Section I Part B, Multiple Choice (graphing calculator required): 13 questions in 38 minutes.
Section II Part A, Free Response (graphing calculator required): 2 questions in 30 minutes.
Section II Part B, Free Response (calculator not permitted): 4 questions in 60 minutes.

Write the derivative you found by the rules, such as $f^{'} (x) = 5 \cos x - \frac{2}{x}$ , before you substitute or use the calculator, so a reader sees the rule and not only the number. Set the calculator to radian mode before any trigonometric work. When a limit has the shape of the definition of the derivative, say which function and which point it is the derivative of before you give its value.

Questions students ask

Why do the sine and cosine rules need radians?

They come from $\lim_{h \to 0} \frac{\sin h}{h} = 1$ , which holds when $h$ is in radians. Measured in degrees, the same limit is $\frac{π}{180}$ , and every slope of sine and cosine picks up that factor.

Why can't I use the power rule on

e^{x}

The power rule is for a variable raised to a fixed number, such as $x^{5}$ . In $e^{x}$ the base is the fixed number and the variable is the exponent, a different kind of function with its own rule: $e^{x}$ is its own derivative.

What is the derivative of

\ln x

x = 0

or at negative

x

There is none. $\ln x$ is defined only for $x > 0$ , and a function has no derivative where it has no value. The rule $\frac{d}{d x} \ln x = \frac{1}{x}$ is for $x > 0$ .

How do I know a limit is a derivative in disguise?

Look for one function evaluated at two inputs on top, one of them a fixed point $a$ , with the difference of those inputs underneath and the limit taken as they come together. A plain number on top may be $f (a)$ in disguise, as $\frac{1}{2}$ is $\cos \frac{π}{3}$ .

Does the minus sign go with the derivative of sine or of cosine?

Cosine: $\frac{d}{d x} \cos x = - \sin x$ . To check, picture the cosine graph just to the right of 0, where it falls from its highest value. A falling graph has a negative slope.

Course alignment, for teachers

AP Calculus AB topic 2.7, Unit 2: Differentiation: Definition and Fundamental Properties.

Skill 1.E: Apply appropriate mathematical rules or procedures, with and without technology.

This lesson maps all 2 essential knowledge statements of the topic to the part that teaches each one: 2 are checked by a scored question. The map is in the lesson file.

What you'll be able to do

Key vocabulary

Part 1 Think first How fast is the buoy moving?

Part 2 Learn it Three short ideas, each checked as you go

7a. Sine and cosine, in radians

7b. The exponential and the natural logarithm

7c. Limits that are derivatives in disguise

Part 3 Watch it done A limit that is a derivative

Part 4 Try it New functions, both ideas

Part 5 Explain it Write it the way the exam asks

Part 6 Check yourself A ride on a Ferris wheel

Part 7 Keep it Bring back earlier ideas

Exam strategy for this topic

Questions students ask

Past exam questions on this topic