AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/2/2-10)
Unit 2 · Topic 2.10
2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
The derivatives of tan x, cot x, sec x and csc x come from writing each one in terms of sine and cosine and using the quotient rule. You should memorize the four results and know where they come from.
Key terms
- derivative of tan x
- derivative of cot x
- derivative of sec x
- derivative of csc x
- trig identities
The four derivatives
Memorize these four. A pattern helps: the three “co” functions (cos, cot, csc) all have a minus sign in their derivatives, tan pairs with sec, and cot pairs with csc.
| Function | Derivative |
|---|---|
| tan x | sec²x |
| cot x | −csc²x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
Deriving tan x
Write tan x = sin x / cos x and use the quotient rule: [cos x·cos x − sin x·(−sin x)] / cos²x = (cos²x + sin²x) / cos²x.
By the Pythagorean identity, cos²x + sin²x = 1. So the derivative is 1/cos²x = sec²x. The derivation for cot x = cos x / sin x is similar and gives −1/sin²x = −csc²x.
Deriving sec x
Write sec x = 1/cos x. Quotient rule: [0·cos x − 1·(−sin x)] / cos²x = sin x / cos²x.
Split it: (1/cos x)·(sin x/cos x) = sec x tan x. Similarly, csc x = 1/sin x gives −cos x / sin²x = −csc x cot x.
Domains and values to know
These derivatives exist wherever the original function is defined. tan x and sec x aren't defined where cos x = 0 (x = π/2 + kπ), and cot x and csc x aren't defined where sin x = 0 (x = kπ).
You'll need exact values at the standard angles. For instance, at x = π/4: tan = 1, sec = √2. At x = π/6: csc = 2, cot = √3. At x = π/3: sec = 2, tan = √3.
Using these with other rules
These derivatives combine with the product and quotient rules just like sin x and cos x. For y = eˣ sec x, the product rule gives y′ = eˣ sec x + eˣ sec x tan x = eˣ sec x(1 + tan x).
Identities can make answers look different but still be equal. Since 1 + tan²x = sec²x, the derivative of tan x can also be written 1 + tan²x. If your answer doesn't match an answer choice, try an identity before deciding you're wrong.
When the angle is something other than x, like tan(3x) or sec(x²), you'll also need the chain rule from 3.1.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Product rule with tan x
Differentiate f(x) = x tan x.
Show the solutionHide the solution
- Step 1: Product rule: f′(x) = (1)(tan x) + x(sec²x).
Answer: f′(x) = tan x + x sec²x
- Example 2
Tangent line to sec x
Find the equation of the line tangent to y = sec x at x = π/4.
Show the solutionHide the solution
- Step 1: Point: sec(π/4) = 1/cos(π/4) = 1/(√2/2) = √2. So the point is (π/4, √2).
- Step 2: Slope: y′ = sec x tan x. At π/4: √2·1 = √2.
- Step 3: Equation: y − √2 = √2(x − π/4).
Answer: y − √2 = √2(x − π/4)
- Example 3
Trap: the missing minus sign
Find the slope of y = csc x at x = π/6.
Show the solutionHide the solution
- Step 1: y′ = −csc x cot x. The minus sign is part of the formula (csc is a “co” function).
- Step 2: csc(π/6) = 1/sin(π/6) = 1/(1/2) = 2, and cot(π/6) = cos(π/6)/sin(π/6) = (√3/2)/(1/2) = √3.
- Step 3: y′(π/6) = −(2)(√3).
- Step 4: Sense check: on (0, π/2), csc x decreases from very large values toward 1, so a negative slope makes sense.
Answer: The slope is −2√3.
Common mistakes
- Dropping the minus sign on the derivatives of cot x and csc x.
- Writing the derivative of sec x as tan²x or sec²x. It's sec x tan x.
- Mixing up sec²x (the derivative of tan x) with 2 sec x.
On the exam
- These derivatives mostly appear in multiple choice, often inside a chain rule or product rule. The exact values at π/6, π/4 and π/3 come up repeatedly.
- The reverse facts will matter in Unit 6: since d/dx tan x = sec²x, an antiderivative of sec²x is tan x.
Connected topics
Videos
Check yourself
4 questions on 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions. Pick an answer to see if you got it, and why.
If y = csc x + cot x, then dy/dx =
If f(x) = sec x, what is f′(π/3) ?
Which of the following is an equation of the line tangent to the graph of y = tan x at x = π/4 ?
If f(x) = x tan x, what is f′(π/4)?
0 of 4 answered