Find $dy/dx$ for the equation below using implicit differentiation.
$xy=\sin\left(y\right)$

• A. $dy/dx=\frac{\sin\left(y\right)}{x}$
• B. $dy/dx=\frac{x}{\cos\left(y\right)}$
• C. $dy/dx=\frac{y}{\cos\left(y\right)-x}$
• D. $dy/dx=\cos\left(y\right)$

Answer: (C)

Verified step by step guidance

1. Start by differentiating both sides of the equation xy = sin(y) with respect to x. Remember that y is a function of x, so you'll need to use implicit differentiation.
2. For the left side, apply the product rule to differentiate xy. The product rule states that d(uv)/dx = u(dv/dx) + v(du/dx). Here, u = x and v = y, so differentiate to get: d(xy)/dx = x(dy/dx) + y.
3. For the right side, differentiate sin(y) with respect to x. Since y is a function of x, use the chain rule: d(sin(y))/dx = cos(y) * (dy/dx).
4. Set the derivatives from both sides equal to each other: x(dy/dx) + y = cos(y) * (dy/dx).
5. Solve for dy/dx by isolating it on one side of the equation. Rearrange the equation to get: dy/dx = (y - x) / (cos(y) - x).