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

Problem

(x+y)3=x3+y3

Solution

1. Differentiate both sides with respect to x using the chain rule for the left side and the power rule for the right side.

d(x+y)/d(x)=d(x3+y3)/d(x)

2. Apply the chain rule to the left side and differentiate term-by-term on the right side, remembering that y is a function of x

3*(x+y)2⋅(1+d(y)/d(x))=3*x2+3*y2d(y)/d(x)

3. Divide by 3 to simplify the equation.

(x+y)2*(1+d(y)/d(x))=x2+y2d(y)/d(x)

4. Expand the left side by distributing (x+y)2

(x+y)2+(x+y)2d(y)/d(x)=x2+y2d(y)/d(x)

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

(x+y)2d(y)/d(x)−y2d(y)/d(x)=x2−(x+y)2

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

d(y)/d(x)*((x+y)2−y2)=x2−(x+y)2

7. Expand the squares to simplify the expressions inside the parentheses.

d(y)/d(x)*(x2+2*x*y+y2−y2)=x2−(x2+2*x*y+y2)

8. Simplify both sides of the equation.

d(y)/d(x)*(x2+2*x*y)=−2*x*y−y2

9. Solve for dy/dx by dividing both sides by (x2+2*x*y)

d(y)/d(x)=(−2*x*y−y2)/(x2+2*x*y)

10. Factor out common variables to reach the simplest form.

d(y)/d(x)=(−y*(2*x+y))/(x*(x+2*y))

Final Answer

d(y)/d(x)=−(2*x*y+y2)/(x2+2*x*y)

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