Find dy/dx square root of xy=x-2y

Problem

√(,x*y)=x−2*y

Solution

1. Rewrite the square root as a power to make differentiation easier.

(x*y)(1/2)=x−2*y

2. Differentiate both sides with respect to x using the chain rule and the product rule on the left side.

d(x*y)/d(x)=d(x−2*y)/d(x)

3. Apply the power rule and product rule to the left side and differentiate the right side term by term.

1/2*(x*y)(−1/2)*(y+xd(y)/d(x))=1−2d(y)/d(x)

4. Distribute the term 1/(2√(,x*y)) to both terms inside the parentheses.

y/(2√(,x*y))+x/(2√(,x*y))d(y)/d(x)=1−2d(y)/d(x)

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

x/(2√(,x*y))d(y)/d(x)+2d(y)/d(x)=1−y/(2√(,x*y))

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

d(y)/d(x)*(x/(2√(,x*y))+2)=1−y/(2√(,x*y))

7. Simplify the expressions on both sides by finding a common denominator.

d(y)/d(x)*((x+4√(,x*y))/(2√(,x*y)))=(2√(,x*y)−y)/(2√(,x*y))

8. Solve for d(y)/d(x) by multiplying both sides by the reciprocal of the coefficient.

d(y)/d(x)=(2√(,x*y)−y)/(x+4√(,x*y))

Final Answer

d(y)/d(x)=(2√(,x*y)−y)/(x+4√(,x*y))

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