Find dy/dx sin(x+y)=xy

Problem

sin(x+y)=x*y

Solution

1. Differentiate both sides with respect to x using the chain rule on the left and the product rule on the right.

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

2. Apply the chain rule to the left side, noting that the derivative of x+y is 1+d(y)/d(x)

cos(x+y)*(1+d(y)/d(x))=(d(x)*y)/d(x)

3. Apply the product rule to the right side.

cos(x+y)*(1+d(y)/d(x))=y+xd(y)/d(x)

4. Distribute the cosine term to separate the terms containing d(y)/d(x)

cos(x+y)+cos(x+y)d(y)/d(x)=y+xd(y)/d(x)

5. Group all terms involving d(y)/d(x) on one side and the remaining terms on the other side.

cos(x+y)d(y)/d(x)−xd(y)/d(x)=y−cos(x+y)

6. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(cos(x+y)−x)=y−cos(x+y)

7. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(y−cos(x+y))/(cos(x+y)−x)

Final Answer

d(y)/d(x)=(y−cos(x+y))/(cos(x+y)−x)

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