Find dy/dx tan(xy)=x+y

Problem

tan(x*y)=x+y

Solution

1. Differentiate both sides with respect to x using the chain rule on the left side and the sum rule on the right side.

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

2. Apply the chain rule to the tangent function, noting that the derivative of tan(u) is sec2(u)

sec2(x*y)⋅d(x*y)/d(x)=1+d(y)/d(x)

3. Apply the product rule to the term x*y inside the derivative.

sec2(x*y)⋅(y+xd(y)/d(x))=1+d(y)/d(x)

4. Distribute the sec2(x*y) term to both parts of the product rule result.

y*sec2(x*y)+x*sec2(x*y)d(y)/d(x)=1+d(y)/d(x)

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

x*sec2(x*y)d(y)/d(x)−d(y)/d(x)=1−y*sec2(x*y)

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

d(y)/d(x)*(x*sec2(x*y)−1)=1−y*sec2(x*y)

7. Solve for the derivative by dividing both sides by the factored expression.

d(y)/d(x)=(1−y*sec2(x*y))/(x*sec2(x*y)−1)

Final Answer

d(y)/d(x)=(1−y*sec2(x*y))/(x*sec2(x*y)−1)

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