Derivative of sec(x)
Secant's derivative looks like something you'd have to memorize, but it comes straight out of writing sec(x) = 1/cos(x) and turning the crank.
Step-by-step solution
sec(x) = 1 / cos(x). Use the quotient rule with u = 1, v = cos(x) (or the reciprocal rule, which is the same thing).
(1/v)′ = −v′/v² gives −(−sin(x)) / cos²(x) = sin(x) / cos²(x).
Split the fraction: [1/cos(x)] · [sin(x)/cos(x)] = sec(x) tan(x).
Why it works
Why bother factoring the answer into sec·tan instead of leaving sin/cos²? Because the factored form reveals structure: the derivative of secant contains secant itself. Functions whose derivatives contain the original function tend to show up in integral tables and differential equations, and sec(x)tan(x) is precisely the integrand that anti-differentiates back to sec(x).
There's also a symmetry worth noticing: the derivatives of the three "co-" functions (cos, cot, csc) all carry minus signs, while sin, tan, and sec don't. If you remember d/dx[sec] = sec·tan, you get d/dx[csc] = −csc·cot for free by swapping every function for its co-version and flipping the sign.
Excluded inputs and slope checks
Sec(x) = 1/cos(x) is defined only where cos(x) ≠ 0: exclude x = π/2 + kπ for every integer k. Its derivative sec(x) tan(x) has the same exclusions. These formulas use radians.
At x = 0, sec(x) = 1 and the slope is zero, so the tangent line is y = 1. At x = π/4, both sec(x) = √2 and tan(x) = 1, so the slope is √2. A positive-looking product formula does not mean the slope is positive everywhere; at x = −π/4 the slope is −√2.
Common mistakes
- Writing tan(x)sec(x) with a minus sign — the negative belongs to cosecant's derivative, not secant's.
- Stopping at sin(x)/cos²(x). This is mathematically equivalent to sec(x)tan(x); follow any requested answer format.
- Confusing sec(x) = 1/cos(x) with cos⁻¹(x), which means arccos — the notation trap that breaks the whole problem before it starts.
Practice problems
Differentiate 2 sec(x)
Answer: 2 sec(x)tan(x).
Find the slope of sec(x) at x = 0
Answer: sec(0)tan(0) = 1·0 = 0 — secant has a flat minimum there.
Differentiate sec(x) + cos(x)
Answer: sec(x)tan(x) − sin(x).