Find dy/dx (2x+2y)^3=8x^3+8y^3

Problem

(2*x+2*y)3=8*x3+8*y3

Solution

1. Expand the left side of the equation using the binomial expansion (a+b)3=a3+3*a2*b+3*a*b2+b3

(2*x+2*y)3=(2*x)3+3*(2*x)2*(2*y)+3*(2*x)*(2*y)2+(2*y)3

(2*x+2*y)3=8*x3+24*x2*y+24*x*y2+8*y3

2. Substitute the expanded form back into the original equation.

8*x3+24*x2*y+24*x*y2+8*y3=8*x3+8*y3

3. Simplify the equation by subtracting 8*x3 and 8*y3 from both sides.

24*x2*y+24*x*y2=0

4. Factor out the common term 24*x*y from the simplified equation.

24*x*y*(x+y)=0

5. Differentiate implicitly with respect to x Since the product 24*x*y*(x+y) equals zero, we apply the product rule to 24*(x2*y+x*y2)=0

d()/d(x)*(x2*y+x*y2)=0

(2*x*y+x2d(y)/d(x))+(y2+2*x*yd(y)/d(x))=0

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

(x2+2*x*y)d(y)/d(x)=−(2*x*y+y2)

7. Solve for d(y)/d(x) by dividing both sides by (x2+2*x*y)

d(y)/d(x)=−(2*x*y+y2)/(x2+2*x*y)

8. Factor the numerator and denominator to simplify the fraction.

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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