For y = -(1/2) sin(3.14/3 - x^2), state the domain and find dy/dx.

Step-by-step solution

1. 1

   State the domain first. The sine function accepts every real input, and $\frac{3.14}{3}-x^{2}$ is a polynomial defined for all $x$, so the composition is defined everywhere:

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

   The number $3.14$ here is a decimal constant (written $3{,}14$ in comma-decimal notation), not $\pi$ — it makes no difference to the differentiation, but it matters if you later solve $y=0$ numerically.

2. 2

   Identify the inner and outer functions. Set

   $u=\frac{3.14}{3}-x^{2},\qquad y=-\frac12\sin u.$

   Naming the inner function explicitly is what keeps the chain rule honest; the constant $-\tfrac12$ simply rides along as a multiplier.

3. 3

   Differentiate the outer function. Since $\frac{d}{du}\sin u=\cos u$,

   $\frac{dy}{du}=-\frac12\cos u.$

   The constant multiple rule lets the $-\tfrac12$ pass straight through the derivative untouched.

4. 4

   Differentiate the inner function and multiply. The constant $\frac{3.14}{3}$ differentiates to $0$, so

   $\frac{du}{dx}=-2x,$

   and by the chain rule

   $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=\left(-\frac12\cos u\right)(-2x)=x\cos\!\left(\frac{3.14}{3}-x^{2}\right).$

   The two minus signs cancel and the $\tfrac12$ cancels against the $2$, which is why the tidy answer has coefficient exactly $1$.

5. 5

   Check numerically and note the critical points. At $x=1.1$ a central difference of $y$ gives $1.0853598$, while $x\cos\!\left(\frac{3.14}{3}-x^{2}\right)=1.0853598$ ✓. The derivative vanishes when $x=0$ or when $\cos\!\left(\frac{3.14}{3}-x^{2}\right)=0$, i.e. $x^{2}=\frac{3.14}{3}-\frac{\pi}{2}-k\pi$ — so $x=0$ is always a critical point, and the others are spaced by the cosine zeros.