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

Problem

y*cos(x)=x2+y2

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the product rule to the left side.

(d(y)*cos(x))/d(x)=d(x2+y2)/d(x)

2. Apply the product rule to y*cos(x) and the power rule along with the chain rule to the right side.

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

3. Simplify the terms by removing parentheses and organizing the expression.

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

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

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

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

d(y)/d(x)*(cos(x)−2*y)=2*x+y*sin(x)

6. Solve for the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(2*x+y*sin(x))/(cos(x)−2*y)

Final Answer

d(y)/d(x)=(2*x+y*sin(x))/(cos(x)−2*y)

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