Find dy/dx square root of xy=9+x^2y

Problem

√(,x*y)=9+x2*y

Solution

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

(x*y)(1/2)=9+x2*y

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

d(x*y)/d(x)=d(9+x2*y)/d(x)

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

1/2*(x*y)(−1/2)⋅(d(x)*y)/d(x)=d(9)/d(x)+(d(x2)*y)/d(x)

4. Apply the product rule to both the x*y and x2*y terms.

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

5. Distribute the terms to isolate the components containing d(y)/d(x)

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

6. Group all terms with 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 factor in 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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