Find dy/dx ysin(x^2)=xsin(y^2)

Problem

y*sin(x2)=x*sin(y2)

Solution

1. Identify the equation as one requiring implicit differentiation because y is not isolated.

2. Differentiate both sides of the equation with respect to x applying the product rule and the chain rule to each side.

3. Apply the product rule to the left side: (d(y)*sin(x2))/d(x)=d(y)/d(x)*sin(x2)+y*cos(x2)⋅2*x

4. Apply the product rule to the right side: (d(x)*sin(y2))/d(x)=1⋅sin(y2)+x*cos(y2)⋅2*yd(y)/d(x)

5. Equate the results of the differentiation:

sin(x2)d(y)/d(x)+2*x*y*cos(x2)=sin(y2)+2*x*y*cos(y2)d(y)/d(x)

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

sin(x2)d(y)/d(x)−2*x*y*cos(y2)d(y)/d(x)=sin(y2)−2*x*y*cos(x2)

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

d(y)/d(x)*(sin(x2)−2*x*y*cos(y2))=sin(y2)−2*x*y*cos(x2)

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

d(y)/d(x)=(sin(y2)−2*x*y*cos(x2))/(sin(x2)−2*x*y*cos(y2))

Final Answer

d(y)/d(x)=(sin(y2)−2*x*y*cos(x2))/(sin(x2)−2*x*y*cos(y2))

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