Find dy/dx (3xy+7)^2=6y

Problem

(3*x*y+7)2=6*y

Solution

1. Differentiate both sides with respect to x using the chain rule on the left side and treating y as a function of x

d(3*x*y+7)/d(x)=(d(6)*y)/d(x)

2. Apply the chain rule to the left side, bringing down the exponent and then differentiating the inner expression.

2*(3*x*y+7)⋅d(3*x*y+7)/d(x)=6d(y)/d(x)

3. Apply the product rule to the term 3*x*y inside the derivative.

2*(3*x*y+7)⋅(3*y+3*xd(y)/d(x))=6d(y)/d(x)

4. Distribute the terms on the left side to isolate the terms containing d(y)/d(x)

(6*x*y+14)⋅(3*y+3*xd(y)/d(x))=6d(y)/d(x)

18*x*y2+18*x2*yd(y)/d(x)+42*y+42*xd(y)/d(x)=6d(y)/d(x)

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

18*x2*yd(y)/d(x)+42*xd(y)/d(x)−6d(y)/d(x)=−18*x*y2−42*y

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

d(y)/d(x)*(18*x2*y+42*x−6)=−18*x*y2−42*y

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

d(y)/d(x)=(−18*x*y2−42*y)/(18*x2*y+42*x−6)

8. Simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 6

d(y)/d(x)=(−3*x*y2−7*y)/(3*x2*y+7*x−1)

Final Answer

d(y)/d(x)=(−3*x*y2−7*y)/(3*x2*y+7*x−1)

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