Find dy/dx 2xy+pisin(y)=2pi

Problem

2*x*y+π*sin(y)=2*π

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule where necessary.

d()/d(x)*(2*x*y+π*sin(y))=d()/d(x)*(2*π)

2. Apply the product rule to the term 2*x*y and the chain rule to the term π*sin(y)

2*y+2*xd(y)/d(x)+π*cos(y)d(y)/d(x)=0

3. Group the terms containing d(y)/d(x) on one side of the equation and move the remaining terms to the other side.

2*xd(y)/d(x)+π*cos(y)d(y)/d(x)=−2*y

4. Factor out the common factor d(y)/d(x) from the left side.

d(y)/d(x)*(2*x+π*cos(y))=−2*y

5. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(−2*y)/(2*x+π*cos(y))

Final Answer

d(y)/d(x)=−(2*y)/(2*x+π*cos(y))

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