Find dy/dx x^3+x^2y+4y^2=6

Problem

x3+x2*y+4*y2=6

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x

d()/d(x)*(x3+x2*y+4*y2)=d()/d(x)*6

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

3*x2+(x2d(y)/d(x)+yd(x2)/d(x))+(d(4)*y2)/d(x)=0

3. Apply the chain rule to the terms involving y

3*x2+x2d(y)/d(x)+2*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 opposite side.

x2d(y)/d(x)+8*yd(y)/d(x)=−3*x2−2*x*y

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

d(y)/d(x)*(x2+8*y)=−3*x2−2*x*y

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

d(y)/d(x)=(−3*x2−2*x*y)/(x2+8*y)

Final Answer

d(y)/d(x)=−(3*x2+2*x*y)/(x2+8*y)

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