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.
Radians and slope examples
The function cos(2x) and its derivative exist for every real x. The argument is measured in radians. The largest absolute slope is 2, compared with 1 for cos(x); this compares maximum slopes, rather than slopes at the same input.
At x = π/4 the function is zero and the derivative is −2 sin(π/2) = −2. Its tangent line is y = −2(x − π/4). At x = 0 the slope is zero, even though the function decreases immediately to the right.
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).