Find Where dy/dx is Equal to Zero x^3+3x^2y+y^3=8

Problem

x3+3*x2*y+y3=8

Solution

1. Differentiate implicitly with respect to x by applying the power rule and the product rule to each term.

d(x3)/d(x)+(d(3)*x2*y)/d(x)+d(y3)/d(x)=d(8)/d(x)

2. Apply the product rule to the term 3*x2*y and the chain rule to y3

3*x2+(6*x*y+3*x2d(y)/d(x))+3*y2d(y)/d(x)=0

3. Isolate the terms containing d(y)/d(x) on one side of the equation.

3*x2d(y)/d(x)+3*y2d(y)/d(x)=−3*x2−6*x*y

4. Factor out d(y)/d(x) and solve for the derivative.

d(y)/d(x)*(3*x2+3*y2)=−3*x2−6*x*y

d(y)/d(x)=(−3*x2−6*x*y)/(3*x2+3*y2)

5. Simplify the expression by dividing the numerator and denominator by 3

d(y)/d(x)=(−x2−2*x*y)/(x2+y2)

6. Set the derivative to zero to find the condition where the slope is horizontal.

(−x2−2*x*y)/(x2+y2)=0

7. Solve for x by setting the numerator equal to zero, assuming the denominator is not zero.

−x2−2*x*y=0

−x*(x+2*y)=0

8. Identify the solutions for the coordinates where the derivative is zero.

x=0

x=−2*y

Final Answer

d(y)/d(x)=0* when *x=0* or *x=−2*y

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