Find dy/dx sin(xy)=y

Problem

sin(x*y)=y

Solution

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

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

2. Apply the chain rule to the left side, which requires using the product rule for the inner term x*y

cos(x*y)⋅(d(x)*y)/d(x)=d(y)/d(x)

3. Apply the product rule to the term x*y where (d(x)*y)/d(x)=y+xd(y)/d(x)

cos(x*y)⋅(y+xd(y)/d(x))=d(y)/d(x)

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

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

5. Isolate the derivative by moving all terms involving d(y)/d(x) to one side of the equation.

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

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

y*cos(x*y)=d(y)/d(x)*(1−x*cos(x*y))

7. Solve for dy/dx by dividing both sides by (1−x*cos(x*y))

d(y)/d(x)=(y*cos(x*y))/(1−x*cos(x*y))

Final Answer

d(y)/d(x)=(y*cos(x*y))/(1−x*cos(x*y))

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