2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Derivatives of tan and cot

Using the quotient rule on sin/cos gives clean results: d/dx[tan x] = sec²x and d/dx[cot x] = -csc²x. Notice the pattern — the 'co' function (cotangent) gets a negative sign, just like cosine's derivative is negative sine. Memorize these as building blocks for the chain rule.

Derivatives of sec and csc

The remaining two: d/dx[sec x] = sec x tan x and d/dx[csc x] = -csc x cot x. Again, the 'co' function (cosecant) picks up the negative sign. Each derivative is the original reciprocal function times its matching 'co-tangent' partner.

Combine with chain rule

These six trig derivatives (sin, cos, tan, cot, sec, csc) are frequently paired with the chain rule when the argument is more complex, like tan(3x²). Always differentiate the outer trig function first using these rules, then multiply by the derivative of the inner function.

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

Pattern for 'co' trig derivatives

Cotangent and cosecant (the 'co' functions) always have negative derivatives, mirroring cosine's negative derivative.

Try an AP-style question

d/dx (tan x) = ?

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