Find dy/dx square root of xy=x^2y+1

Problem

√(,x*y)=x2*y+1

Solution

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

(x*y)(1/2)=x2*y+1

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

d(x*y)/d(x)=d(x2*y+1)/d(x)

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

1/2*(x*y)(−1/2)*(xd(y)/d(x)+y)=d(x2*y+1)/d(x)

4. Apply the product rule to the right side.

1/(2√(,x*y))*(xd(y)/d(x)+y)=x2d(y)/d(x)+2*x*y

5. Distribute the terms on the left side to isolate the derivative.

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

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

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

7. 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))

8. 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)

9. 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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