Unit 2Derivatives of

\tan x

\cot x

\sec x

\csc x

Concept 4.193Built for May 2027 AP Exam

Derivatives of $\tan x$ , $\cot x$ , $\sec x$ , $\csc x$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Four formulas complete the trigonometric set: $\frac{d}{dx}\tan x = \sec^2 x$ , $\frac{d}{dx}\cot x = -\csc^2 x$ , $\frac{d}{dx}\sec x = \sec x \tan x$ , and $\frac{d}{dx}\csc x = -\csc x \cot x$ . The signs are not random. Both co-functions carry a minus, and the two that remain positive do not. Each formula pairs a function with a companion: the tangent's derivative is a secant squared, and the secant's derivative is a secant times a tangent. None of the four needs to be memorised if you can rewrite the function in sines and cosines and apply the quotient rule, and that fallback is available on every problem that uses them.

On the AP Exam

What you’ll see: These appear far more often as an intermediate step than as the object of a question: a tangent slope on a trigonometric curve, a rate in a context, or an answer option that differs from the correct one only in sign.
What you’ll do: Produce the derivative, check the sign against the co- rule, and keep the answer in the same function family the question used.
Memory hook: Co- means minus. Cotangent and cosecant take the minus; tangent and secant do not.
Use it on the exam: For MCQs: When two options differ only in sign, the co- rule decides it in one second. When they differ in the second factor, ask which companion function pairs with the one you differentiated. For FRQs: A derivative that is not simplified to match a target form usually costs the point even when the values agree. If the question states an expression to reach, convert your answer toward it rather than stopping at the first correct form.

Concept Review

Four Formulas, One Sign Rule

Figure 1 — The three co-family results and the sign rule that produces two of them. The amber cards are the ones that carry a minus, and each corresponds to a name beginning with co-.

Four cards giving the derivatives of cotangent, secant, cosecant, and the natural logarithm, with the two negative results highlighted

The Four Formulas

Function	Derivative	Sign	Domain restriction
$\tan x$	$\sec^2 x$	positive	$\cos x \neq 0$
$\cot x$	$-\csc^2 x$	negative	$\sin x \neq 0$
$\sec x$	$\sec x \tan x$	positive	$\cos x \neq 0$
$\csc x$	$-\csc x \cot x$	negative	$\sin x \neq 0$

The pattern is worth stating as a rule rather than a list. A name beginning with co- produced by swapping sine and cosine inherits a minus, because the swap reverses the direction of the quotient. Names that are not co-functions carry no minus.

Where the Formulas Come From

Each one follows from the quotient rule applied to a ratio of sine and cosine:

$\frac{d}{dx}\tan x = \frac{d}{dx}\frac{\sin x}{\cos x} = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

The numerator collapses to $\cos^2 x + \sin^2 x = 1$ , which is the Pythagorean identity doing the work. The cotangent is the same computation with the roles reversed, which is where its minus comes from. Reconstructing one formula from scratch is a reliable way to check a remembered one.

The Fallback That Always Works

If a formula does not come back under exam pressure, rewrite the function:

Function	Rewrite	Derivative route
$\tan x$	$\frac{\sin x}{\cos x}$	Quotient rule
$\cot x$	$\frac{\cos x}{\sin x}$	Quotient rule
$\sec x$	$\frac{1}{\cos x}$	Quotient rule
$\csc x$	$\frac{1}{\sin x}$	Quotient rule

Every one of the four is a quotient of two functions whose derivatives you already know, so the fallback costs about thirty seconds and never fails. That matters more than memorisation, because a remembered formula with the wrong sign is worse than a derived one with the right sign.

Common Misconception Check

Myth 1: "The derivative of $\sec x$ is $\sec^2 x$ ." Truth: That is the tangent's derivative. The secant's derivative is $\sec x \tan x$ , a product rather than a square.
Myth 2: "The derivative of $\cot x$ is $\csc^2 x$ ." Truth: It is $-\csc^2 x$ . Cotangent begins with co-, so it takes the minus.
Myth 3: "These formulas hold for every real $x$ ." Truth: All four are undefined where their denominator vanishes, and the derivative lives on the same restricted domain as the function.

Topic-Specific Study Method

Sign First, Then Formula

Borrowed idea: Four functions, four formulas, and one sign rule that generates half of them. Every name beginning with co- carries a minus in its derivative. The tangent and secant pair up, since differentiating one produces the other times a companion. When a formula refuses to return, rewrite the function in sines and cosines and use the quotient rule. That route always works and never needs to be remembered under pressure. Check the domain too, because all four vanish wherever the cosine or the sine does. Write the answer in the same family the question used.

You are asked for	Sign	Formula	Companion
$\frac{d}{dx}\tan x$	plus	$\sec^2 x$	secant squared
$\frac{d}{dx}\cot x$	minus	$\csc^2 x$	cosecant squared
$\frac{d}{dx}\sec x$	plus	$\sec x \tan x$	secant times tangent
$\frac{d}{dx}\csc x$	minus	$\csc x \cot x$	cosecant times cotangent

① Derive all four from the quotient rule in a single sitting. After the second one, the pattern stops needing to be memorised.

② Cover the table above and write the four derivatives from the sign rule alone, then check. The signs should come back before the formulas do, which is the correct order of recall.

③ Differentiate $\tan x$ twice and confirm that the second derivative, $2\sec^2 x \tan x$ , is positive wherever the tangent is increasing. Consistency checks like this catch a lost square.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = \tan x + \sec x$ .

Standard Execution:

Term by term. $\frac{d}{dx}\tan x = \sec^2 x$ and $\frac{d}{dx}\sec x = \sec x \tan x$ .

Assemble. $f'(x) = \sec^2 x + \sec x \tan x$ .

Factor if useful. $f'(x) = \sec x(\sec x + \tan x)$ .

Check a value. At $x = 0$ this gives $1(1 + 0) = 1$ , and the graph of $\tan x + \sec x$ is rising at slope 1 there.

Exam Variation (Context / Parameter Shift): A quantity is modeled by $Q(t) = 3\cot t$ for $0 < t < \pi$ . Find $Q'(t)$ and determine whether $Q$ is increasing on that interval.

Solution: $Q'(t) = -3\csc^2 t$ . Since $\csc^2 t > 0$ wherever it is defined, $Q'(t) < 0$ throughout $0 < t < \pi$ , so $Q$ is decreasing and never increasing. The minus from the co- rule is what settles the direction.

Common Traps & Failure Signals:

The missing minus (Signal: a positive derivative for a decreasing co-function): the sign rule was skipped.
The wrong companion (Signal: $\sec^2 x$ written for the secant): the tangent's formula was borrowed.
The missing square (Signal: $\sec x$ where $\sec^2 x$ belongs): the degree of the answer does not match the tangent's growth.

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

Sign Skipped: You dropped the minus on a co-function.
Companion Swapped $\rightarrow$ Name the function before naming its partner.
Formula Borrowed: You used the tangent's result for the secant, which happens when the four are stored as a block of similar-looking expressions instead of as two pairs with opposite signs.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; These derivatives appear as intermediate steps inside multiple-choice items; no released BC free-response part from 2017 to 2026 scores them as a separate object
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; All 54 free-response questions from the nine released BC sets were read part by part
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

The Tangent

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

MCQ 2

The Cosecant

If $g(x) = \csc x$ , what is $g'(x)$ ?

MCQ 3

The Sign Rule

Which of these derivatives does not carry a built-in minus from the co- rule?

FRQ 1 • A Trigonometric Model

A quantity is modeled by $Q(t) = 3\cot t + 2\sec t$ for $0 < t < \pi/2$ , measured in liters, with $t$ in hours.

(a) Find $Q'(t)$ .

(b) Determine whether $Q$ is increasing or decreasing on $0 < t < \pi/2$ , and justify your answer.

(c) Find $Q'(\pi/4)$ exactly.

(d) A classmate writes $Q'(t) = 3\csc^2 t + 2\sec t \tan t$ and concludes that $Q$ is increasing. Identify the error and explain how a sign error changes the conclusion.

Cheatsheet

tan, cot, sec, csc

	Derivative	Sign
$\tan x$	$\sec^2 x$	plus
$\cot x$	$-\csc^2 x$	minus
$\sec x$	$\sec x \tan x$	plus
$\csc x$	$-\csc x \cot x$	minus
Fallback	Rewrite as sine over cosine	Quotient rule

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Start Topic Practice Next Topic: 3.1 Chain Rule Formula

Unit 2Derivatives of

\tan x

\cot x

\sec x

\csc x

Concept 4.193Built for May 2027 AP Exam

Derivatives of $\tan x$ , $\cot x$ , $\sec x$ , $\csc x$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: These appear far more often as an intermediate step than as the object of a question: a tangent slope on a trigonometric curve, a rate in a context, or an answer option that differs from the correct one only in sign.
What you’ll do: Produce the derivative, check the sign against the co- rule, and keep the answer in the same function family the question used.
Memory hook: Co- means minus. Cotangent and cosecant take the minus; tangent and secant do not.
Use it on the exam: For MCQs: When two options differ only in sign, the co- rule decides it in one second. When they differ in the second factor, ask which companion function pairs with the one you differentiated. For FRQs: A derivative that is not simplified to match a target form usually costs the point even when the values agree. If the question states an expression to reach, convert your answer toward it rather than stopping at the first correct form.

Concept Review

Four Formulas, One Sign Rule

Figure 1 — The three co-family results and the sign rule that produces two of them. The amber cards are the ones that carry a minus, and each corresponds to a name beginning with co-.

Four cards giving the derivatives of cotangent, secant, cosecant, and the natural logarithm, with the two negative results highlighted

The Four Formulas

Function	Derivative	Sign	Domain restriction
$\tan x$	$\sec^2 x$	positive	$\cos x \neq 0$
$\cot x$	$-\csc^2 x$	negative	$\sin x \neq 0$
$\sec x$	$\sec x \tan x$	positive	$\cos x \neq 0$
$\csc x$	$-\csc x \cot x$	negative	$\sin x \neq 0$

Where the Formulas Come From

Each one follows from the quotient rule applied to a ratio of sine and cosine:

$\frac{d}{dx}\tan x = \frac{d}{dx}\frac{\sin x}{\cos x} = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

The Fallback That Always Works

If a formula does not come back under exam pressure, rewrite the function:

Function	Rewrite	Derivative route
$\tan x$	$\frac{\sin x}{\cos x}$	Quotient rule
$\cot x$	$\frac{\cos x}{\sin x}$	Quotient rule
$\sec x$	$\frac{1}{\cos x}$	Quotient rule
$\csc x$	$\frac{1}{\sin x}$	Quotient rule

Common Misconception Check

Myth 1: "The derivative of $\sec x$ is $\sec^2 x$ ." Truth: That is the tangent's derivative. The secant's derivative is $\sec x \tan x$ , a product rather than a square.
Myth 2: "The derivative of $\cot x$ is $\csc^2 x$ ." Truth: It is $-\csc^2 x$ . Cotangent begins with co-, so it takes the minus.
Myth 3: "These formulas hold for every real $x$ ." Truth: All four are undefined where their denominator vanishes, and the derivative lives on the same restricted domain as the function.

Topic-Specific Study Method

Sign First, Then Formula

You are asked for	Sign	Formula	Companion
$\frac{d}{dx}\tan x$	plus	$\sec^2 x$	secant squared
$\frac{d}{dx}\cot x$	minus	$\csc^2 x$	cosecant squared
$\frac{d}{dx}\sec x$	plus	$\sec x \tan x$	secant times tangent
$\frac{d}{dx}\csc x$	minus	$\csc x \cot x$	cosecant times cotangent

① Derive all four from the quotient rule in a single sitting. After the second one, the pattern stops needing to be memorised.

② Cover the table above and write the four derivatives from the sign rule alone, then check. The signs should come back before the formulas do, which is the correct order of recall.

③ Differentiate $\tan x$ twice and confirm that the second derivative, $2\sec^2 x \tan x$ , is positive wherever the tangent is increasing. Consistency checks like this catch a lost square.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = \tan x + \sec x$ .

Standard Execution:

Term by term. $\frac{d}{dx}\tan x = \sec^2 x$ and $\frac{d}{dx}\sec x = \sec x \tan x$ .

Assemble. $f'(x) = \sec^2 x + \sec x \tan x$ .

Factor if useful. $f'(x) = \sec x(\sec x + \tan x)$ .

Check a value. At $x = 0$ this gives $1(1 + 0) = 1$ , and the graph of $\tan x + \sec x$ is rising at slope 1 there.

Exam Variation (Context / Parameter Shift): A quantity is modeled by $Q(t) = 3\cot t$ for $0 < t < \pi$ . Find $Q'(t)$ and determine whether $Q$ is increasing on that interval.

Solution: $Q'(t) = -3\csc^2 t$ . Since $\csc^2 t > 0$ wherever it is defined, $Q'(t) < 0$ throughout $0 < t < \pi$ , so $Q$ is decreasing and never increasing. The minus from the co- rule is what settles the direction.

Common Traps & Failure Signals:

The missing minus (Signal: a positive derivative for a decreasing co-function): the sign rule was skipped.
The wrong companion (Signal: $\sec^2 x$ written for the secant): the tangent's formula was borrowed.
The missing square (Signal: $\sec x$ where $\sec^2 x$ belongs): the degree of the answer does not match the tangent's growth.

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

Sign Skipped: You dropped the minus on a co-function.
Companion Swapped $\rightarrow$ Name the function before naming its partner.
Formula Borrowed: You used the tangent's result for the secant, which happens when the four are stored as a block of similar-looking expressions instead of as two pairs with opposite signs.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; These derivatives appear as intermediate steps inside multiple-choice items; no released BC free-response part from 2017 to 2026 scores them as a separate object
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; All 54 free-response questions from the nine released BC sets were read part by part
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

The Tangent

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

MCQ 2

The Cosecant

If $g(x) = \csc x$ , what is $g'(x)$ ?

MCQ 3

The Sign Rule

Which of these derivatives does not carry a built-in minus from the co- rule?

FRQ 1 • A Trigonometric Model

A quantity is modeled by $Q(t) = 3\cot t + 2\sec t$ for $0 < t < \pi/2$ , measured in liters, with $t$ in hours.

(a) Find $Q'(t)$ .

(b) Determine whether $Q$ is increasing or decreasing on $0 < t < \pi/2$ , and justify your answer.

(c) Find $Q'(\pi/4)$ exactly.

(d) A classmate writes $Q'(t) = 3\csc^2 t + 2\sec t \tan t$ and concludes that $Q$ is increasing. Identify the error and explain how a sign error changes the conclusion.

Cheatsheet

tan, cot, sec, csc

	Derivative	Sign
$\tan x$	$\sec^2 x$	plus
$\cot x$	$-\csc^2 x$	minus
$\sec x$	$\sec x \tan x$	plus
$\csc x$	$-\csc x \cot x$	minus
Fallback	Rewrite as sine over cosine	Quotient rule

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

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

Start Topic Practice Next Topic: 3.1 Chain Rule Formula

Unit 2Derivatives of

\tan x

\cot x

\sec x

\csc x

Concept 4.193Built for May 2027 AP Exam

Derivatives of $\tan x$ , $\cot x$ , $\sec x$ , $\csc x$

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

What you’ll see: These appear far more often as an intermediate step than as the object of a question: a tangent slope on a trigonometric curve, a rate in a context, or an answer option that differs from the correct one only in sign.
What you’ll do: Produce the derivative, check the sign against the co- rule, and keep the answer in the same function family the question used.
Memory hook: Co- means minus. Cotangent and cosecant take the minus; tangent and secant do not.
Use it on the exam: For MCQs: When two options differ only in sign, the co- rule decides it in one second. When they differ in the second factor, ask which companion function pairs with the one you differentiated. For FRQs: A derivative that is not simplified to match a target form usually costs the point even when the values agree. If the question states an expression to reach, convert your answer toward it rather than stopping at the first correct form.

Concept Review

Four Formulas, One Sign Rule

Figure 1 — The three co-family results and the sign rule that produces two of them. The amber cards are the ones that carry a minus, and each corresponds to a name beginning with co-.

Four cards giving the derivatives of cotangent, secant, cosecant, and the natural logarithm, with the two negative results highlighted

The Four Formulas

Function	Derivative	Sign	Domain restriction
$\tan x$	$\sec^2 x$	positive	$\cos x \neq 0$
$\cot x$	$-\csc^2 x$	negative	$\sin x \neq 0$
$\sec x$	$\sec x \tan x$	positive	$\cos x \neq 0$
$\csc x$	$-\csc x \cot x$	negative	$\sin x \neq 0$

Where the Formulas Come From

Each one follows from the quotient rule applied to a ratio of sine and cosine:

$\frac{d}{dx}\tan x = \frac{d}{dx}\frac{\sin x}{\cos x} = \frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$

The Fallback That Always Works

If a formula does not come back under exam pressure, rewrite the function:

Function	Rewrite	Derivative route
$\tan x$	$\frac{\sin x}{\cos x}$	Quotient rule
$\cot x$	$\frac{\cos x}{\sin x}$	Quotient rule
$\sec x$	$\frac{1}{\cos x}$	Quotient rule
$\csc x$	$\frac{1}{\sin x}$	Quotient rule

Common Misconception Check

Myth 1: "The derivative of $\sec x$ is $\sec^2 x$ ." Truth: That is the tangent's derivative. The secant's derivative is $\sec x \tan x$ , a product rather than a square.
Myth 2: "The derivative of $\cot x$ is $\csc^2 x$ ." Truth: It is $-\csc^2 x$ . Cotangent begins with co-, so it takes the minus.
Myth 3: "These formulas hold for every real $x$ ." Truth: All four are undefined where their denominator vanishes, and the derivative lives on the same restricted domain as the function.

Topic-Specific Study Method

Sign First, Then Formula

You are asked for	Sign	Formula	Companion
$\frac{d}{dx}\tan x$	plus	$\sec^2 x$	secant squared
$\frac{d}{dx}\cot x$	minus	$\csc^2 x$	cosecant squared
$\frac{d}{dx}\sec x$	plus	$\sec x \tan x$	secant times tangent
$\frac{d}{dx}\csc x$	minus	$\csc x \cot x$	cosecant times cotangent

① Derive all four from the quotient rule in a single sitting. After the second one, the pattern stops needing to be memorised.

② Cover the table above and write the four derivatives from the sign rule alone, then check. The signs should come back before the formulas do, which is the correct order of recall.

③ Differentiate $\tan x$ twice and confirm that the second derivative, $2\sec^2 x \tan x$ , is positive wherever the tangent is increasing. Consistency checks like this catch a lost square.

Training & Error Diagnosis

Core Archetype (Simplest Form): Differentiate $f(x) = \tan x + \sec x$ .

Standard Execution:

Term by term. $\frac{d}{dx}\tan x = \sec^2 x$ and $\frac{d}{dx}\sec x = \sec x \tan x$ .

Assemble. $f'(x) = \sec^2 x + \sec x \tan x$ .

Factor if useful. $f'(x) = \sec x(\sec x + \tan x)$ .

Check a value. At $x = 0$ this gives $1(1 + 0) = 1$ , and the graph of $\tan x + \sec x$ is rising at slope 1 there.

Exam Variation (Context / Parameter Shift): A quantity is modeled by $Q(t) = 3\cot t$ for $0 < t < \pi$ . Find $Q'(t)$ and determine whether $Q$ is increasing on that interval.

Solution: $Q'(t) = -3\csc^2 t$ . Since $\csc^2 t > 0$ wherever it is defined, $Q'(t) < 0$ throughout $0 < t < \pi$ , so $Q$ is decreasing and never increasing. The minus from the co- rule is what settles the direction.

Common Traps & Failure Signals:

The missing minus (Signal: a positive derivative for a decreasing co-function): the sign rule was skipped.
The wrong companion (Signal: $\sec^2 x$ written for the secant): the tangent's formula was borrowed.
The missing square (Signal: $\sec x$ where $\sec^2 x$ belongs): the degree of the answer does not match the tangent's growth.

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

Sign Skipped: You dropped the minus on a co-function.
Companion Swapped $\rightarrow$ Name the function before naming its partner.
Formula Borrowed: You used the tangent's result for the secant, which happens when the four are stored as a block of similar-looking expressions instead of as two pairs with opposite signs.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; These derivatives appear as intermediate steps inside multiple-choice items; no released BC free-response part from 2017 to 2026 scores them as a separate object
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; All 54 free-response questions from the nine released BC sets were read part by part
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

The Tangent

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

MCQ 2

The Cosecant

If $g(x) = \csc x$ , what is $g'(x)$ ?

MCQ 3

The Sign Rule

Which of these derivatives does not carry a built-in minus from the co- rule?

FRQ 1 • A Trigonometric Model

A quantity is modeled by $Q(t) = 3\cot t + 2\sec t$ for $0 < t < \pi/2$ , measured in liters, with $t$ in hours.

(a) Find $Q'(t)$ .

(b) Determine whether $Q$ is increasing or decreasing on $0 < t < \pi/2$ , and justify your answer.

(c) Find $Q'(\pi/4)$ exactly.

(d) A classmate writes $Q'(t) = 3\csc^2 t + 2\sec t \tan t$ and concludes that $Q$ is increasing. Identify the error and explain how a sign error changes the conclusion.

Cheatsheet

tan, cot, sec, csc

	Derivative	Sign
$\tan x$	$\sec^2 x$	plus
$\cot x$	$-\csc^2 x$	minus
$\sec x$	$\sec x \tan x$	plus
$\csc x$	$-\csc x \cot x$	minus
Fallback	Rewrite as sine over cosine	Quotient rule

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