find dy/dx by implicit differentiation. 4 cos x sin y = 2 y =

Explanation:

Step1: Differentiate both sides

Differentiate $4\cos x\sin y$ and $2$ with respect to $x$. Using the product - rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = 4\cos x$ and $v=\sin y$. The derivative of the left - hand side is $4(-\sin x)\sin y+4\cos x\cos y\frac{dy}{dx}$, and the derivative of the right - hand side is $0$. So, $- 4\sin x\sin y+4\cos x\cos y\frac{dy}{dx}=0$.

Step2: Solve for $\frac{dy}{dx}$

First, isolate the term with $\frac{dy}{dx}$: $4\cos x\cos y\frac{dy}{dx}=4\sin x\sin y$. Then divide both sides by $4\cos x\cos y$ (assuming $\cos x\cos y\neq0$) to get $\frac{dy}{dx}=\frac{\sin x\sin y}{\cos x\cos y}$. Using the tangent formula $\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$, we can rewrite it as $\frac{dy}{dx}=\tan x\tan y$.

Answer:

$\tan x\tan y$