Find dy/dx 4x^2+xy+4y^2=9

Problem

4*x2+x*y+4*y2=9

Solution

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

(d(4)*x2)/d(x)+(d(x)*y)/d(x)+(d(4)*y2)/d(x)=d(9)/d(x)

2. Apply the power rule to the first term and the product rule to the second term.

8*x+(xd(y)/d(x)+y⋅1)+(d(4)*y2)/d(x)=0

3. Apply the chain rule to the term involving y2

8*x+xd(y)/d(x)+y+8*yd(y)/d(x)=0

4. Group the terms containing d(y)/d(x) on one side and move all other terms to the other side.

xd(y)/d(x)+8*yd(y)/d(x)=−8*x−y

5. Factor out the common factor of d(y)/d(x)

d(y)/d(x)*(x+8*y)=−8*x−y

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

d(y)/d(x)=(−8*x−y)/(x+8*y)

Final Answer

d(y)/d(x)=−(8*x+y)/(x+8*y)

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