Find dy/dx xy=cot(xy)

Problem

x*y=cot(x*y)

Solution

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

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

2. Apply the product rule to the left side and the chain rule to the right side, noting that the derivative of cot(u) is −csc2(u)

xd(y)/d(x)+y=−csc2(x*y)(d(x)*y)/d(x)

3. Expand the derivative on the right side using the product rule again.

xd(y)/d(x)+y=−csc2(x*y)*(xd(y)/d(x)+y)

4. Distribute the −csc2(x*y) term to the terms inside the parentheses.

xd(y)/d(x)+y=−x*csc2(x*y)d(y)/d(x)−y*csc2(x*y)

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

xd(y)/d(x)+x*csc2(x*y)d(y)/d(x)=−y*csc2(x*y)−y

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

d(y)/d(x)*(x+x*csc2(x*y))=−y*csc2(x*y)−y

7. Solve for dy/dx by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(−y*(csc2(x*y)+1))/(x*(1+csc2(x*y)))

8. Simplify the fraction by canceling the common factor (1+csc2(x*y)) from the numerator and the denominator.

d(y)/d(x)=−y/x

Final Answer

d(y)/d(x)=−y/x

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