Find dy/dx x^4(x+y)=y^2(3x-y)

Problem

x4*(x+y)=y2*(3*x−y)

Solution

1. Expand both sides of the equation to prepare for implicit differentiation.

x5+x4*y=3*x*y2−y3

2. Differentiate both sides with respect to x applying the product rule to terms containing both x and y and the chain rule to terms containing y

d(x5)/d(x)+(d(x4)*y)/d(x)=(d(3)*x*y2)/d(x)−d(y3)/d(x)

3. Apply the differentiation rules to each term.

5*x4+(4*x3*y+x4d(y)/d(x))=(3*y2+6*x*yd(y)/d(x))−3*y2d(y)/d(x)

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

x4d(y)/d(x)−6*x*yd(y)/d(x)+3*y2d(y)/d(x)=3*y2−5*x4−4*x3*y

5. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(x4−6*x*y+3*y2)=3*y2−5*x4−4*x3*y

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

d(y)/d(x)=(3*y2−5*x4−4*x3*y)/(x4−6*x*y+3*y2)

Final Answer

d(y)/d(x)=(3*y2−5*x4−4*x3*y)/(x4−6*x*y+3*y2)

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