Find dy/dx 3x+y^3-(4y)/(x+2)=10x^2

Problem

3*x+y3−(4*y)/(x+2)=10*x2

Solution

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

d()/d(x)*(3*x+y3−(4*y)/(x+2))=d()/d(x)*(10*x2)

2. Apply the sum rule and differentiate the first two terms on the left side.

3+3*y2d(y)/d(x)−d()/d(x)(4*y)/(x+2)=20*x

3. Apply the quotient rule to the term (4*y)/(x+2) where u=4*y and v=x+2

d()/d(x)(4*y)/(x+2)=((x+2)*(4d(y)/d(x))−(4*y)*(1))/((x+2)2)

4. Substitute the quotient rule result back into the main equation.

3+3*y2d(y)/d(x)−(4*(x+2)d(y)/d(x)−4*y)/((x+2)2)=20*x

5. Multiply the entire equation by (x+2)2 to clear the denominator.

3*(x+2)2+3*y2*(x+2)2d(y)/d(x)−(4*(x+2)d(y)/d(x)−4*y)=20*x*(x+2)2

6. Distribute the negative sign and group all terms containing d(y)/d(x) on one side.

3*y2*(x+2)2d(y)/d(x)−4*(x+2)d(y)/d(x)=20*x*(x+2)2−3*(x+2)2−4*y

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

d(y)/d(x)*(3*y2*(x+2)2−4*(x+2))=20*x*(x+2)2−3*(x+2)2−4*y

8. Solve for d(y)/d(x) by dividing both sides by the factored expression.

d(y)/d(x)=(20*x*(x+2)2−3*(x+2)2−4*y)/(3*y2*(x+2)2−4*(x+2))

Final Answer

d(y)/d(x)=((20*x−3)*(x+2)2−4*y)/(3*y2*(x+2)2−4*(x+2))

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