2. find the derivative of the following function. simplify as much as possible. show all of your work…

Explanation:

Step1: Use the cosine - difference formula

The left - hand side is $\cos(x - y)$ by the formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$. So the equation becomes $\cos(x - y)=\frac{1}{2}$.

Step2: Differentiate both sides with respect to $x$

Differentiating $\cos(x - y)$ with respect to $x$ using the chain rule. Let $u=x - y$, then $\frac{d}{dx}\cos(u)=-\sin(u)\cdot(1-\frac{dy}{dx})$. The derivative of the right - hand side $\frac{d}{dx}(\frac{1}{2}) = 0$. So we have $-\sin(x - y)\cdot(1-\frac{dy}{dx})=0$.

Step3: Solve for $\frac{dy}{dx}$

If $\sin(x - y)\neq0$, then $1-\frac{dy}{dx}=0$, and $\frac{dy}{dx}=1$. If $\sin(x - y)=0$, the equation $-\sin(x - y)\cdot(1 - \frac{dy}{dx})=0$ is satisfied for all values of $\frac{dy}{dx}$. But from $\cos(x - y)=\frac{1}{2}$, we know that $x - y = 2k\pi\pm\frac{\pi}{3},k\in\mathbb{Z}$, and $\sin(x - y)\neq0$ in general for the non - degenerate case. So $\frac{dy}{dx}=1$.

Answer:

$\frac{dy}{dx}=1$