Find dy/dx square root of xy=4+x^2y

Problem

√(,x*y)=4+x2*y

Solution

1. Rewrite the square root as a power to prepare for differentiation.

√(,x*y)=(x*y)(1/2)

2. Differentiate both sides of the equation with respect to x using the chain rule and the product rule.

d(x*y)/d(x)=d(4+x2*y)/d(x)

3. Apply the chain rule and product rule to the left side.

1/2*(x*y)(−1/2)*(y+xd(y)/d(x))=0+2*x*y+x2d(y)/d(x)

4. Distribute the term on the left side to isolate the derivative terms.

y/(2√(,x*y))+x/(2√(,x*y))d(y)/d(x)=2*x*y+x2d(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)−x2d(y)/d(x)=2*x*y−y/(2√(,x*y))

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

d(y)/d(x)*(x/(2√(,x*y))−x2)=2*x*y−y/(2√(,x*y))

7. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(2*x*y−y/(2√(,x*y)))/(x/(2√(,x*y))−x2)

8. Simplify the complex fraction by multiplying the numerator and denominator by 2√(,x*y)

d(y)/d(x)=(4*x*y√(,x*y)−y)/(x−2*x2√(,x*y))

Final Answer

d(y)/d(x)=(4*x*y√(,x*y)−y)/(x−2*x2√(,x*y))

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