Derivative of tan(2x)
tan(2x) combines the standard tangent derivative with a chain-rule factor — a two-ingredient problem that previews how every composite trig derivative works.
Step-by-step solution
Outer function tan(u) with derivative sec²(u); inner function u = 2x with derivative 2.
The outer derivative with the inside intact is sec²(2x).
f′(x) = 2 sec²(2x).
Why it works
Compressing tangent horizontally by a factor of 2 doesn't just double its slope — it also doubles how often the function blows up. tan(2x) has vertical asymptotes every π/2 units instead of every π, and its derivative 2 sec²(2x) inherits them exactly: the derivative explodes precisely where the function does, at odd multiples of π/4.
Since sec² is never less than 1, the derivative 2 sec²(2x) is never less than 2. Every branch of tan(2x) therefore climbs at least twice as fast as the line y = x at every single point — a concrete way to feel how violent the compressed tangent's growth really is between its asymptotes.
Common mistakes
- Forgetting the leading 2 from the chain rule.
- Writing sec²(x) — the argument of the answer must remain 2x, matching the original inside.
- Squaring incorrectly as 2 sec(2x)² read as 2[sec(2x)]... the notation sec²(u) means [sec(u)]², applied before any other operations.
Practice problems
Differentiate tan(3x)
Answer: 3 sec²(3x).
Differentiate tan(2x) − 2x
Answer: 2 sec²(2x) − 2 = 2 tan²(2x), using sec² − 1 = tan².
Find the slope of tan(2x) at x = 0
Answer: 2 sec²(0) = 2.