Unit 2Derivatives of Trigonometric and Exponential/Log Functions

Concept 4.141Built for May 2027 AP Exam

Derivatives of Trigonometric and Exponential/Log Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Six derivatives carry this section. The sine goes to cosine, the cosine goes to negative sine, and the tangent goes to $\sec^2 x$ . The natural exponential is its own derivative, the natural logarithm gives $\frac{1}{x}$ , and a general base gives $a^x \ln a$ . Two features separate these from the power rule. First, the trigonometric pairs cycle rather than shrink, so differentiating four times returns the original function. Second, the exponential and the logarithm are inverse processes of each other, which is why one produces itself and the other produces a reciprocal. The single most common error is not the formulas themselves but the argument: everything here is stated for $x$ , and a compound argument needs the chain rule on top.

On the AP Exam

What you’ll see: A trigonometric or exponential model whose rate, tangent, or second derivative is requested, usually inside a particle, temperature, or population context.
What you’ll do: Differentiate the outer function, then check whether the argument is more than plain $x$ before moving on.
Memory hook: Cycle, self, reciprocal. Trig cycles, the natural exponential reproduces itself, and the logarithm returns a reciprocal.
Use it on the exam: For MCQs: The distractors flip a minus sign or swap sine and cosine in the cycle. Differentiate twice and check that the second derivative of $\sin$ is $-\sin$ , which catches both errors at once. For FRQs: When a rate is supplied rather than computed, the derivative rule still matters for the second derivative. Parts that ask about concavity usually need you to differentiate a rate that was handed to you in the previous line.

Concept Review

Six Derivatives, One Cycle

Figure 1 — The three trigonometric results on the top row and the three exponential and logarithmic results on the bottom. The amber card is the only trigonometric derivative that carries a minus, and that minus is what makes the sine-cosine pair cycle instead of repeat.

Six derivative cards for sine, cosine, tangent, the natural exponential, the natural logarithm, and a general base exponential, with the one negative result highlighted

The Trigonometric Cycle

Function	First derivative	Second	Third	Fourth
$\sin x$	$\cos x$	$-\sin x$	$-\cos x$	$\sin x$
$\cos x$	$-\sin x$	$-\cos x$	$\sin x$	$\cos x$

Four derivatives return the original function, and the pattern is why trigonometric functions describe oscillation: a quantity whose second derivative is its own negative is being pulled back toward equilibrium. The minus in the cosine's derivative is not decoration; remove it and the cycle stops closing.

Why the Exponential Reproduces Itself

The derivative of $e^x$ is $e^x$ because the limit

$\lim_{h \to 0}\frac{e^{x+h}-e^x}{h} = e^x \lim_{h\to 0}\frac{e^h - 1}{h} = e^x \cdot 1$

exists with the constant equal to one. That constant is what makes $e$ special: for any other base the same limit equals $\ln a$ , so

$\frac{d}{dx}a^x = a^x \ln a$

Base $e$ is the fixed point of the family, the base whose growth rate matches its value exactly. Every exponential model on the exam is either written in base $e$ or converted into it for exactly this reason.

The Logarithm and the Inverse Relationship

Since $\ln x$ undoes $e^x$ , its derivative undoes the self-reproduction: $\frac{d}{dx}\ln x = \frac{1}{x}$ . The pair is worth holding as one fact rather than two. A quantity growing exponentially has a rate proportional to itself; a quantity growing logarithmically has a rate inversely proportional to its input. Both statements come from the same inverse relationship.

Common Misconception Check

Myth 1: "The derivative of $\cos x$ is $\sin x$ ." Truth: It is $-\sin x$ . Without the minus, four derivatives would return $\cos x$ with the wrong sign and the cycle would not close.
Myth 2: "The derivative of $2^x$ is $2^x$ ." Truth: Only base $e$ reproduces itself. For base 2 the derivative is $2^x \ln 2$ , which is smaller than $2^x$ because $\ln 2 < 1$ .
Myth 3: "The derivative of $\ln x$ is $\frac{1}{x}$ for every nonzero $x$ ." Truth: The domain of $\ln x$ is $x > 0$ , and the derivative lives on the same domain.

Topic-Specific Study Method

Name the Pair

Borrowed idea: Name the pair. Sine goes to cosine, cosine goes to negative sine, and the minus is the only thing separating the two directions. The exponential keeps its shape under differentiation, which is why every model built on $e^x$ has a rate proportional to its own value, and the logarithm is the inverse that turns that proportionality into a division by the input. Watch the argument, not just the function, because the most common error here is differentiating an inside that is not plain $x$ .

You see	First derivative	Watch for
$\sin x$	$\cos x$	nothing to add
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	square on the secant
$e^x$	$e^x$	base $e$ only
$2^x$	$2^x \ln 2$	the logarithm factor
$\ln x$	$\frac{1}{x}$	domain $x > 0$

① Differentiate $\sin x$ four times and write the four results in a column. The return to the start is the cycle, and it is easier to trust after seeing it once.

② Sketch $e^x$ and $2^x$ on the same axes, then compare each curve's slope at $x = 0$ with its height. For $e^x$ the two numbers agree; for $2^x$ the slope is about 69 percent of the height, which is $\ln 2$ .

③ Take every trigonometric derivative in this section and write its sign pattern as a row of plus and minus marks. The row is the fastest way to check a remembered formula.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the slope of the line tangent to $f(x) = e^x + \cos x$ at $x = 0$ .

Standard Execution:

Differentiate. $f'(x) = e^x - \sin x$ .

Substitute. $f'(0) = 1 - 0 = 1$ .

Check the point. $f(0) = 1 + 1 = 2$ .

Report. The tangent line is $y - 2 = x$ , so the slope is $1$ .

Exam Variation (Context / Parameter Shift): A particle moves along a line with position $s(t) = 3\sin t + 2\cos t$ for $t \ge 0$ , in meters and seconds. Find the acceleration at $t = \pi/2$ and determine whether the particle is speeding up or slowing down there.

Solution: $v(t) = 3\cos t - 2\sin t$ and $a(t) = -3\sin t - 2\cos t$ . At $t = \pi/2$ this gives $v = -2$ and $a = -3$ . The velocity and acceleration share a sign, so the speed is increasing and the particle is speeding up.

Common Traps & Failure Signals:

The missing minus (Signal: $\cos x$ differentiated to $+\sin x$ ): the cycle was memorised as a rotation without its sign.
The base confusion (Signal: $2^x$ differentiated to $2^x$ ): the special case of $e$ was generalised to every base.
The unexamined argument (Signal: $e^{3x}$ differentiated to $e^{3x}$ ): the inner function was ignored, which the chain rule repairs.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Sign Dropped: You lost the minus on a cosine derivative.
Base Assumed $\rightarrow$ Check the base before copying the self-reproducing result.
Argument Ignored: You differentiated the outer function and stopped there, which is correct only when the argument is exactly $x$ and wrong the moment a coefficient or a sum appears inside it.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; A released part asks for the derivative of an explicitly given model in five of the nine sets, and the family involved is almost always a trig, exponential, or log function
10-Year Total Count: 5 of 9 Exams · 6 FRQ Parts; Counted where the derivative of the given family is scored, not where a calculator supplies the number
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2021 (Q2a), 2023 (Q1d, Q2a), 2026 (Q4A)

Practice Examples

MCQ 1

The Cycle

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

MCQ 2

A General Base

If $g(x) = 3^x$ , what is $g'(x)$ ?

MCQ 3

The Logarithm

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

FRQ 1 • A Damped Oscillation

A weight on a spring oscillates so that its height above the floor is $y(t) = 40 + 6\cos(2t)$ centimeters for $t \ge 0$ seconds.

(a) Find $y'(t)$ , the rate at which the height is changing.

(b) Find $y'(0)$ and $y''(0)$ , and explain what the signs say about the motion at $t = 0$ .

(c) Find every time $t$ in $0 \le t \le \pi$ at which the weight is momentarily at rest.

(d) A student claims that because the height never falls below 34 centimeters, the rate of change can never be negative. Identify the error.

Cheatsheet

Trig and Exp/Log

	Derivative	Note
$\sin x$	$\cos x$	starts the cycle
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	squared secant
$e^x$	$e^x$	fixed point
$a^x$	$a^x \ln a$	general base
$\ln x$	$\frac{1}{x}$	$x > 0$

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 2.13 Product Rule Formula

Unit 2Derivatives of Trigonometric and Exponential/Log Functions

Concept 4.141Built for May 2027 AP Exam

Derivatives of Trigonometric and Exponential/Log Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A trigonometric or exponential model whose rate, tangent, or second derivative is requested, usually inside a particle, temperature, or population context.
What you’ll do: Differentiate the outer function, then check whether the argument is more than plain $x$ before moving on.
Memory hook: Cycle, self, reciprocal. Trig cycles, the natural exponential reproduces itself, and the logarithm returns a reciprocal.
Use it on the exam: For MCQs: The distractors flip a minus sign or swap sine and cosine in the cycle. Differentiate twice and check that the second derivative of $\sin$ is $-\sin$ , which catches both errors at once. For FRQs: When a rate is supplied rather than computed, the derivative rule still matters for the second derivative. Parts that ask about concavity usually need you to differentiate a rate that was handed to you in the previous line.

Concept Review

Six Derivatives, One Cycle

Six derivative cards for sine, cosine, tangent, the natural exponential, the natural logarithm, and a general base exponential, with the one negative result highlighted

The Trigonometric Cycle

Function	First derivative	Second	Third	Fourth
$\sin x$	$\cos x$	$-\sin x$	$-\cos x$	$\sin x$
$\cos x$	$-\sin x$	$-\cos x$	$\sin x$	$\cos x$

Why the Exponential Reproduces Itself

The derivative of $e^x$ is $e^x$ because the limit

$\lim_{h \to 0}\frac{e^{x+h}-e^x}{h} = e^x \lim_{h\to 0}\frac{e^h - 1}{h} = e^x \cdot 1$

exists with the constant equal to one. That constant is what makes $e$ special: for any other base the same limit equals $\ln a$ , so

$\frac{d}{dx}a^x = a^x \ln a$

The Logarithm and the Inverse Relationship

Common Misconception Check

Myth 1: "The derivative of $\cos x$ is $\sin x$ ." Truth: It is $-\sin x$ . Without the minus, four derivatives would return $\cos x$ with the wrong sign and the cycle would not close.
Myth 2: "The derivative of $2^x$ is $2^x$ ." Truth: Only base $e$ reproduces itself. For base 2 the derivative is $2^x \ln 2$ , which is smaller than $2^x$ because $\ln 2 < 1$ .
Myth 3: "The derivative of $\ln x$ is $\frac{1}{x}$ for every nonzero $x$ ." Truth: The domain of $\ln x$ is $x > 0$ , and the derivative lives on the same domain.

Topic-Specific Study Method

Name the Pair

You see	First derivative	Watch for
$\sin x$	$\cos x$	nothing to add
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	square on the secant
$e^x$	$e^x$	base $e$ only
$2^x$	$2^x \ln 2$	the logarithm factor
$\ln x$	$\frac{1}{x}$	domain $x > 0$

① Differentiate $\sin x$ four times and write the four results in a column. The return to the start is the cycle, and it is easier to trust after seeing it once.

③ Take every trigonometric derivative in this section and write its sign pattern as a row of plus and minus marks. The row is the fastest way to check a remembered formula.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the slope of the line tangent to $f(x) = e^x + \cos x$ at $x = 0$ .

Standard Execution:

Differentiate. $f'(x) = e^x - \sin x$ .

Substitute. $f'(0) = 1 - 0 = 1$ .

Check the point. $f(0) = 1 + 1 = 2$ .

Report. The tangent line is $y - 2 = x$ , so the slope is $1$ .

Solution: $v(t) = 3\cos t - 2\sin t$ and $a(t) = -3\sin t - 2\cos t$ . At $t = \pi/2$ this gives $v = -2$ and $a = -3$ . The velocity and acceleration share a sign, so the speed is increasing and the particle is speeding up.

Common Traps & Failure Signals:

The missing minus (Signal: $\cos x$ differentiated to $+\sin x$ ): the cycle was memorised as a rotation without its sign.
The base confusion (Signal: $2^x$ differentiated to $2^x$ ): the special case of $e$ was generalised to every base.
The unexamined argument (Signal: $e^{3x}$ differentiated to $e^{3x}$ ): the inner function was ignored, which the chain rule repairs.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Sign Dropped: You lost the minus on a cosine derivative.
Base Assumed $\rightarrow$ Check the base before copying the self-reproducing result.
Argument Ignored: You differentiated the outer function and stopped there, which is correct only when the argument is exactly $x$ and wrong the moment a coefficient or a sum appears inside it.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; A released part asks for the derivative of an explicitly given model in five of the nine sets, and the family involved is almost always a trig, exponential, or log function
10-Year Total Count: 5 of 9 Exams · 6 FRQ Parts; Counted where the derivative of the given family is scored, not where a calculator supplies the number
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2021 (Q2a), 2023 (Q1d, Q2a), 2026 (Q4A)

Practice Examples

MCQ 1

The Cycle

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

MCQ 2

A General Base

If $g(x) = 3^x$ , what is $g'(x)$ ?

MCQ 3

The Logarithm

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

FRQ 1 • A Damped Oscillation

A weight on a spring oscillates so that its height above the floor is $y(t) = 40 + 6\cos(2t)$ centimeters for $t \ge 0$ seconds.

(a) Find $y'(t)$ , the rate at which the height is changing.

(b) Find $y'(0)$ and $y''(0)$ , and explain what the signs say about the motion at $t = 0$ .

(c) Find every time $t$ in $0 \le t \le \pi$ at which the weight is momentarily at rest.

(d) A student claims that because the height never falls below 34 centimeters, the rate of change can never be negative. Identify the error.

Cheatsheet

Trig and Exp/Log

	Derivative	Note
$\sin x$	$\cos x$	starts the cycle
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	squared secant
$e^x$	$e^x$	fixed point
$a^x$	$a^x \ln a$	general base
$\ln x$	$\frac{1}{x}$	$x > 0$

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 2.13 Product Rule Formula

Unit 2Derivatives of Trigonometric and Exponential/Log Functions

Concept 4.141Built for May 2027 AP Exam

Derivatives of Trigonometric and Exponential/Log Functions

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: A trigonometric or exponential model whose rate, tangent, or second derivative is requested, usually inside a particle, temperature, or population context.
What you’ll do: Differentiate the outer function, then check whether the argument is more than plain $x$ before moving on.
Memory hook: Cycle, self, reciprocal. Trig cycles, the natural exponential reproduces itself, and the logarithm returns a reciprocal.
Use it on the exam: For MCQs: The distractors flip a minus sign or swap sine and cosine in the cycle. Differentiate twice and check that the second derivative of $\sin$ is $-\sin$ , which catches both errors at once. For FRQs: When a rate is supplied rather than computed, the derivative rule still matters for the second derivative. Parts that ask about concavity usually need you to differentiate a rate that was handed to you in the previous line.

Concept Review

Six Derivatives, One Cycle

Six derivative cards for sine, cosine, tangent, the natural exponential, the natural logarithm, and a general base exponential, with the one negative result highlighted

The Trigonometric Cycle

Function	First derivative	Second	Third	Fourth
$\sin x$	$\cos x$	$-\sin x$	$-\cos x$	$\sin x$
$\cos x$	$-\sin x$	$-\cos x$	$\sin x$	$\cos x$

Why the Exponential Reproduces Itself

The derivative of $e^x$ is $e^x$ because the limit

$\lim_{h \to 0}\frac{e^{x+h}-e^x}{h} = e^x \lim_{h\to 0}\frac{e^h - 1}{h} = e^x \cdot 1$

exists with the constant equal to one. That constant is what makes $e$ special: for any other base the same limit equals $\ln a$ , so

$\frac{d}{dx}a^x = a^x \ln a$

The Logarithm and the Inverse Relationship

Common Misconception Check

Myth 1: "The derivative of $\cos x$ is $\sin x$ ." Truth: It is $-\sin x$ . Without the minus, four derivatives would return $\cos x$ with the wrong sign and the cycle would not close.
Myth 2: "The derivative of $2^x$ is $2^x$ ." Truth: Only base $e$ reproduces itself. For base 2 the derivative is $2^x \ln 2$ , which is smaller than $2^x$ because $\ln 2 < 1$ .
Myth 3: "The derivative of $\ln x$ is $\frac{1}{x}$ for every nonzero $x$ ." Truth: The domain of $\ln x$ is $x > 0$ , and the derivative lives on the same domain.

Topic-Specific Study Method

Name the Pair

You see	First derivative	Watch for
$\sin x$	$\cos x$	nothing to add
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	square on the secant
$e^x$	$e^x$	base $e$ only
$2^x$	$2^x \ln 2$	the logarithm factor
$\ln x$	$\frac{1}{x}$	domain $x > 0$

① Differentiate $\sin x$ four times and write the four results in a column. The return to the start is the cycle, and it is easier to trust after seeing it once.

③ Take every trigonometric derivative in this section and write its sign pattern as a row of plus and minus marks. The row is the fastest way to check a remembered formula.

Training & Error Diagnosis

Core Archetype (Simplest Form): Find the slope of the line tangent to $f(x) = e^x + \cos x$ at $x = 0$ .

Standard Execution:

Differentiate. $f'(x) = e^x - \sin x$ .

Substitute. $f'(0) = 1 - 0 = 1$ .

Check the point. $f(0) = 1 + 1 = 2$ .

Report. The tangent line is $y - 2 = x$ , so the slope is $1$ .

Solution: $v(t) = 3\cos t - 2\sin t$ and $a(t) = -3\sin t - 2\cos t$ . At $t = \pi/2$ this gives $v = -2$ and $a = -3$ . The velocity and acceleration share a sign, so the speed is increasing and the particle is speeding up.

Common Traps & Failure Signals:

The missing minus (Signal: $\cos x$ differentiated to $+\sin x$ ): the cycle was memorised as a rotation without its sign.
The base confusion (Signal: $2^x$ differentiated to $2^x$ ): the special case of $e$ was generalised to every base.
The unexamined argument (Signal: $e^{3x}$ differentiated to $e^{3x}$ ): the inner function was ignored, which the chain rule repairs.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Sign Dropped: You lost the minus on a cosine derivative.
Base Assumed $\rightarrow$ Check the base before copying the self-reproducing result.
Argument Ignored: You differentiated the outer function and stopped there, which is correct only when the argument is exactly $x$ and wrong the moment a coefficient or a sum appears inside it.

Past 10-Year AP Exam Appearance

Testing Frequency: Most Years; A released part asks for the derivative of an explicitly given model in five of the nine sets, and the family involved is almost always a trig, exponential, or log function
10-Year Total Count: 5 of 9 Exams · 6 FRQ Parts; Counted where the derivative of the given family is scored, not where a calculator supplies the number
FRQ Years Appeared: 2018 (Q2a), 2019 (Q1d), 2021 (Q2a), 2023 (Q1d, Q2a), 2026 (Q4A)

Practice Examples

MCQ 1

The Cycle

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

MCQ 2

A General Base

If $g(x) = 3^x$ , what is $g'(x)$ ?

MCQ 3

The Logarithm

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

FRQ 1 • A Damped Oscillation

A weight on a spring oscillates so that its height above the floor is $y(t) = 40 + 6\cos(2t)$ centimeters for $t \ge 0$ seconds.

(a) Find $y'(t)$ , the rate at which the height is changing.

(b) Find $y'(0)$ and $y''(0)$ , and explain what the signs say about the motion at $t = 0$ .

(c) Find every time $t$ in $0 \le t \le \pi$ at which the weight is momentarily at rest.

(d) A student claims that because the height never falls below 34 centimeters, the rate of change can never be negative. Identify the error.

Cheatsheet

Trig and Exp/Log

	Derivative	Note
$\sin x$	$\cos x$	starts the cycle
$\cos x$	$-\sin x$	the minus
$\tan x$	$\sec^2 x$	squared secant
$e^x$	$e^x$	fixed point
$a^x$	$a^x \ln a$	general base
$\ln x$	$\frac{1}{x}$	$x > 0$

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 2.13 Product Rule Formula