Derivative of cos(2x)
cos(2x) stacks the two most-forgotten details in beginner differentiation — cosine's minus sign and the chain rule's inner factor — into a single problem.
Step-by-step solution
Outer function cosine, inner function u = 2x. Chain rule required.
d/du[cos(u)] = −sin(u), so the outer derivative is −sin(2x) — minus sign included, inside untouched.
The inside 2x differentiates to 2: f′(x) = −2 sin(2x).
Why it works
Both pieces of the answer carry independent meaning. The minus sign says cos(2x) starts at its peak (value 1 at x = 0) and immediately falls, so its slope must go negative first. The 2 says it falls twice as fast as ordinary cosine, because the doubled frequency compresses the whole wave horizontally without changing its height.
This function is also a quiet workhorse of trig identities: cos(2x) equals 1 − 2sin²(x), and differentiating that form with the chain rule gives −4 sin(x)cos(x) = −2 sin(2x) — the same answer by a completely different route. Cross-checking derivatives through identities like this is one of the best habits a calculus student can build.
Common mistakes
- Losing the minus sign (answering 2 sin(2x)) or losing the 2 (answering −sin(2x)) — each error kills half the answer.
- Changing the argument to sin(x).
- Double-negating: the second derivative is −4 cos(2x), and sign errors compound quickly across repeated differentiation.
Practice problems
Differentiate cos(3x)
Answer: −3 sin(3x).
Differentiate 5 cos(2x)
Answer: −10 sin(2x).
Find the second derivative of cos(2x)
Answer: −4 cos(2x).