For y = 2cos(x), state the domain and find the derivative.

Step-by-step solution

1. 1

   Determine the domain. The cosine function is defined for every real input — there is no division, no even root and no logarithm to restrict it — and multiplying by $2$ changes nothing. Hence

   $\mathcal{D}=\mathbb{R}$

2. 2

   Pull the constant out. The constant multiple rule says $\dfrac{d}{dx}\left[cf(x)\right]=c\,f'(x)$, so the $2$ simply waits outside:

   $y'=2\cdot\frac{d}{dx}\left(\cos x\right)$

3. 3

   Differentiate the cosine. The standard derivative carries a minus sign:

   $\frac{d}{dx}\cos x=-\sin x$

   Losing this minus sign is the most common error with trigonometric derivatives. (The argument is plain $x$, so no chain rule factor is needed here.)

4. 4

   Combine.

   $y'=2(-\sin x)=-2\sin x$

5. 5

   Verify numerically and structurally. A central difference with $h=10^{-6}$ matches $-2\sin x$ at $x=0.3$, $1.4$ and $2.9$ to within $10^{-6}$ ✓. Structural check: $y=2\cos x$ has maxima at $x=0,2\pi,\ldots$ and minima at $x=\pi,\ldots$, and indeed $y'=-2\sin x$ vanishes exactly at every multiple of $\pi$ ✓. The range of $y'$ is $[-2,2]$, so the curve's steepest slopes are $\pm2$, attained at $x=\tfrac{\pi}{2}$ and $\tfrac{3\pi}{2}$.