Find dy/dx x^2+6y^2+6y=20

Problem

x2+6*y2+6*y=20

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule to terms involving y

d(x2)/d(x)+(d(6)*y2)/d(x)+(d(6)*y)/d(x)=d(20)/d(x)

2. Apply differentiation rules to each term, using the power rule for x2 and the chain rule for terms with y

2*x+12*yd(y)/d(x)+6d(y)/d(x)=0

3. Isolate the terms containing d(y)/d(x) on one side of the equation by subtracting 2*x from both sides.

12*yd(y)/d(x)+6d(y)/d(x)=−2*x

4. Factor out the common factor of d(y)/d(x) from the left side.

d(y)/d(x)*(12*y+6)=−2*x

5. Solve for the derivative by dividing both sides by (12*y+6)

d(y)/d(x)=(−2*x)/(12*y+6)

6. Simplify the fraction by dividing the numerator and the denominator by their greatest common factor, which is 2

d(y)/d(x)=(−x)/(6*y+3)

Final Answer

d(y)/d(x)=−x/(6*y+3)

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