Derivative of cos(x)
Cosine's derivative is almost as tidy as sine's — with one minus sign that students forget more than any other in first-year calculus.
Step-by-step solution
You are differentiating f(x) = cos(x) — the argument is plain x, so the standard derivative applies directly.
From the limit definition (or from the identity cos(x) = sin(π/2 − x) plus the chain rule), the derivative of cosine is negative sine.
f′(x) = −sin(x).
Why it works
The minus sign has a physical meaning, not just an algebraic one. Cosine starts at its maximum value of 1 when x = 0 and immediately decreases. The slope is negative just to the right of zero — and indeed −sin(x) is negative just after 0. If you ever blank on which trig derivative carries the minus, sketch one hump of cosine and ask whether it's rising or falling at the start.
Sine and cosine are each other's derivatives up to sign, which is what makes the pair the fundamental solutions of the equation f″ = −f — the equation governing springs, pendulums, and every simple oscillation in physics. Differentiating cos(x) twice gives −cos(x): the function reproduces its own negative, which is oscillation in a nutshell.
Domain and tangent line
Cos(x) is differentiable for every real x. Standard trigonometric derivative formulas use radians; for a degree-valued input t, differentiate cos(πt/180) to obtain −(π/180) sin(πt/180).
At x = π/2 the point is (π/2, 0) and the slope is −1, giving the tangent line y = −(x − π/2). At x = 0 the slope is zero. Being decreasing on an interval does not require a strictly negative derivative at every point.
Common mistakes
- Dropping the negative sign and writing sin(x). This is the classic error — the minus belongs to cosine's derivative.
- Doubling the negative for the second derivative and getting +cos(x) too early: the chain is cos → −sin → −cos → sin → cos.
- Confusing d/dx[cos(x)] with the integral: ∫cos(x)dx = sin(x) + C has no minus sign, and the two answers are easy to swap under exam pressure.
Practice problems
Find the derivative of 3 cos(x) − 2
Answer: −3 sin(x) — the constant −2 vanishes.
Find the slope of cos(x) at x = π/2
Answer: −sin(π/2) = −1: cosine is falling at its steepest there.
What's the 3rd derivative of cos(x)?
Answer: sin(x) — follow the cycle cos → −sin → −cos → sin.