Short on time? This page is the recap. For the derivations, the graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: sec²x means (sec x)². Angles are in radians.
Recap
- tan, cot, sec and csc are all quotients of sin x and cos x. Rewrite, then use the quotient rule.
- The Pythagorean identity sin²x + cos²x = 1 tidies the numerator for tan and cot.
- The “co” functions (cot, csc) get a minus sign, just as cos x does.
- Each derivative exists exactly where the original function is defined.
- sec²x ≥ 1, so tan x is increasing on each interval of its domain, with slope at least 1. −csc²x ≤ −1, so cot x is decreasing on each interval of its domain.
Key relationships
| Function | Rewrite | Derivative | Not defined at |
|---|---|---|---|
| tan x | sin x / cos x | sec²x | x = π/2 + kπ |
| cot x | cos x / sin x | −csc²x | x = kπ |
| sec x | 1 / cos x | sec x tan x | x = π/2 + kπ |
| csc x | 1 / sin x | −csc x cot x | x = kπ |
Useful identities: sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = csc²x.
Exact values you will need: tan(π/4) = 1, sec(π/4) = √2, sec(π/3) = 2, tan(π/3) = √3, csc(π/6) = 2, cot(π/6) = √3.
Assumptions behind the results
- x is measured in radians.
- x is in the domain of the function (the denominator sin x or cos x is not 0).
- For a product or quotient that contains these functions, the product or quotient rule is still needed. (Compositions such as tan(3x) need the chain rule from Topic 3.1.)
Mistakes to avoid
- d/dx tan x = sec x (missing square).
- d/dx sec x = tan x (missing the factor sec x).
- Dropping the minus sign on cot x or csc x.
- Mixing up the reciprocals: sec x = 1/cos x, csc x = 1/sin x.
- Treating sin⁻¹x as 1/sin x.
- Multiplying derivatives in a product such as x sec x.
Quick self-check
- Find the slope of y = tan x at x = π/3. (sec²(π/3) = 2² = 4)
- Find the derivative of csc x at x = π/2. (−csc(π/2)cot(π/2) = −1 × 0 = 0)
- Differentiate sec x + cot x, and evaluate at π/4. (sec x tan x − csc²x; at π/4: √2 × 1 − 2 = √2 − 2)
Next: practice questions.