Final answer to the problem
$\left(-y-xy^{\prime}\right)\sin\left(xy\right)=2+2y\cdot y^{\prime}$
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Step-by-step Solution
Learn how to solve sum rule of differentiation problems step by step online. Find the implicit derivative d/dx(cos(xy)=2x+y^2). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if f(x) = \cos(x), then f'(x) = -\sin(x)\cdot D_x(x). Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x and g=y. The derivative of the linear function is equal to 1.
Final answer to the problem
$\left(-y-xy^{\prime}\right)\sin\left(xy\right)=2+2y\cdot y^{\prime}$
