Find dy/dx (4x+4y)^3=64x^3+64y^3

Problem

(4*x+4*y)^3=64*x^3+64*y^3

Solution

1. Simplify the equation by factoring out the constant 4 from the left side.

(4*(x+y))^3=64*x^3+64*y^3

64*(x+y)^3=64*x^3+64*y^3

2. Divide both sides by 64 to isolate the binomial cube.

(x+y)^3=x^3+y^3

3. Expand the left side using the binomial expansion formula (a+b)^3=a^3+3*a^2*b+3*a*b^2+b^3

x^3+3*x^2*y+3*x*y^2+y^3=x^3+y^3

4. Subtract x^3 and y^3 from both sides to simplify the equation.

3*x^2*y+3*x*y^2=0

5. Factor the resulting expression.

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

6. Differentiate implicitly with respect to x Since the product equals zero, we can differentiate the simplified form 3*x^2*y+3*x*y^2=0 using the product rule.

(d(3)*x^2*y)/d(x)+(d(3)*x*y^2)/d(x)=0

(6*x*y+3*x^2d(y)/d(x))+(3*y^2+6*x*yd(y)/d(x))=0

7. Group the terms containing d(y)/d(x) on one side.

(3*x^2+6*x*y)d(y)/d(x)=−6*x*y−3*y^2

8. Solve for d(y)/d(x) by dividing.

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

d(y)/d(x)=−(y*(2*x+y))/(x*(x+2*y))

Final Answer

d(y)/d(x)=−(2*x*y+y^2)/(x^2+2*x*y)

Want more problems? Check here!