2.10 Derivatives of tan, cot, sec, csc

The four

d/dx[tan x] = sec²x, d/dx[cot x] = −csc²x, d/dx[sec x] = sec x·tan x, d/dx[csc x] = −csc x·cot x.

The pattern

The 'co-' functions (cos, cot, csc) all differentiate with a negative sign. tan and sec stay positive.

Key terms

d/dx tan x

= sec²x.

d/dx cot x

= −csc²x.

d/dx sec x

= sec x·tan x.

d/dx csc x

= −csc x·cot x.

Sign pattern

Co-functions (cos, cot, csc) differentiate with a negative sign.

Try an AP-style question

d/dx (tan x) = ?

1. sec² x
2. −csc² x
3. sec x tan x
4. cot x