Find dy/dx xy^3+xy=8

Problem

x*y3+x*y=8

Solution

1. Differentiate both sides with respect to x using implicit differentiation.

d()/d(x)*(x*y3+x*y)=d(8)/d(x)

2. Apply the product rule to the first term x*y3 where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

(d(x)*y3)/d(x)=x⋅3*y2d(y)/d(x)+y3⋅1

3. Apply the product rule to the second term x*y

(d(x)*y)/d(x)=xd(y)/d(x)+y⋅1

4. Combine the results and set the derivative of the constant 8 to 0

3*x*y2d(y)/d(x)+y3+xd(y)/d(x)+y=0

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

3*x*y2d(y)/d(x)+xd(y)/d(x)=−y3−y

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

d(y)/d(x)*(3*x*y2+x)=−y3−y

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

d(y)/d(x)=(−y3−y)/(3*x*y2+x)

Final Answer

d(y)/d(x)=(−y3−y)/(3*x*y2+x)

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