Find dy/dx sin(x+y)=y^2cos(x)

Problem

sin(x+y)=y2*cos(x)

Solution

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

d(sin(x+y))/d(x)=(d(y2)*cos(x))/d(x)

2. Apply the chain rule to the left side and the product rule to the right side.

cos(x+y)*(1+d(y)/d(x))=2*yd(y)/d(x)*cos(x)+y2*(−sin(x))

3. Distribute the cosine term on the left side to isolate the derivative terms.

cos(x+y)+cos(x+y)d(y)/d(x)=2*y*cos(x)d(y)/d(x)−y2*sin(x)

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

cos(x+y)d(y)/d(x)−2*y*cos(x)d(y)/d(x)=−y2*sin(x)−cos(x+y)

5. Factor out d(y)/d(x) from the left side of the equation.

d(y)/d(x)*(cos(x+y)−2*y*cos(x))=−(y2*sin(x)+cos(x+y))

6. Solve for the derivative by dividing both sides by the factored expression.

d(y)/d(x)=(−(y2*sin(x)+cos(x+y)))/(cos(x+y)−2*y*cos(x))

7. Simplify the signs by multiplying the numerator and denominator by −1

d(y)/d(x)=(y2*sin(x)+cos(x+y))/(2*y*cos(x)−cos(x+y))

Final Answer

d(y)/d(x)=(y2*sin(x)+cos(x+y))/(2*y*cos(x)−cos(x+y))

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