Find dy/dx xsin(y)=ycos(x)

Problem

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

Solution

1. Differentiate both sides with respect to x using the product rule and the chain rule for terms involving y

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

2. Apply the product rule to the left side, where d(sin(y))/d(x)=cos(y)d(y)/d(x)

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

3. Apply the product rule to the right side, where d(cos(x))/d(x)=−sin(x)

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

4. Rearrange the equation to group all terms containing d(y)/d(x) on one side and the remaining terms on the other.

x*cos(y)d(y)/d(x)−cos(x)d(y)/d(x)=−y*sin(x)−sin(y)

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

d(y)/d(x)*(x*cos(y)−cos(x))=−y*sin(x)−sin(y)

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

d(y)/d(x)=(−y*sin(x)−sin(y))/(x*cos(y)−cos(x))

7. Simplify the expression by multiplying the numerator and denominator by −1 to clean up the signs.

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

Final Answer

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

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