Find dy/dx ycos(x)=1+sin(xy)

Problem

y*cos(x)=1+sin(x*y)

Solution

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

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

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

d(y)/d(x)*cos(x)+y*(−sin(x))=0+cos(x*y)(d(x)*y)/d(x)

3. Apply the product rule to the term (d(x)*y)/d(x) on the right side.

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

4. Distribute the cos(x*y) term on the right side.

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

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

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

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

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

7. Solve for d(y)/d(x) by dividing both sides by the factor (cos(x)−x*cos(x*y))

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

Final Answer

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

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