9. use implicit differentiation to find dy/dx. y² = x² + sin(xy)

Explanation:

Step1: Differentiate both sides

Differentiate (y^{2}=x^{2}+\sin(xy)) with respect to (x). Using the chain - rule, (\frac{d}{dx}(y^{2}) = 2y\frac{dy}{dx}). Using the power - rule, (\frac{d}{dx}(x^{2})=2x). For (\frac{d}{dx}(\sin(xy))), use the chain - rule and product - rule. Let (u = xy), then (\frac{d}{dx}(\sin(u))=\cos(u)\cdot\frac{d}{dx}(xy)). By the product - rule (\frac{d}{dx}(xy)=y + x\frac{dy}{dx}). So (\frac{d}{dx}(\sin(xy))=\cos(xy)\left(y + x\frac{dy}{dx}\right)). The differentiated equation is (2y\frac{dy}{dx}=2x+\cos(xy)\left(y + x\frac{dy}{dx}\right)).

Step2: Expand and collect terms with (\frac{dy}{dx})

Expand the right - hand side: (2y\frac{dy}{dx}=2x + y\cos(xy)+x\cos(xy)\frac{dy}{dx}). Move the terms with (\frac{dy}{dx}) to one side: (2y\frac{dy}{dx}-x\cos(xy)\frac{dy}{dx}=2x + y\cos(xy)). Factor out (\frac{dy}{dx}): (\frac{dy}{dx}(2y - x\cos(xy))=2x + y\cos(xy)).

Step3: Solve for (\frac{dy}{dx})

Divide both sides by ((2y - x\cos(xy))) to get (\frac{dy}{dx}=\frac{2x + y\cos(xy)}{2y - x\cos(xy)}).

Answer:

(\frac{dy}{dx}=\frac{2x + y\cos(xy)}{2y - x\cos(xy)})