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

Problem

x*y4+x2*y=x+3*y

Solution

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

d(x*y4+x2*y)/d(x)=d(x+3*y)/d(x)

2. Apply the product rule to the terms on the left side.

(y4+x⋅4*y3d(y)/d(x))+(2*x*y+x2d(y)/d(x))=1+3d(y)/d(x)

3. Group all terms containing d(y)/d(x) on one side of the equation and all other terms on the opposite side.

4*x*y3d(y)/d(x)+x2d(y)/d(x)−3d(y)/d(x)=1−y4−2*x*y

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

d(y)/d(x)*(4*x*y3+x2−3)=1−y4−2*x*y

5. Solve for the derivative by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(1−y4−2*x*y)/(4*x*y3+x2−3)

Final Answer

d(y)/d(x)=(1−y4−2*x*y)/(4*x*y3+x2−3)

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