Find dy/dx x^3+y^3=8

Problem

x3+y3=8

Solution

1. Differentiate both sides of the equation with respect to x treating y as a function of x

d(x3)/d(x)+d(y3)/d(x)=d(8)/d(x)

2. Apply the power rule to the x term and the chain rule to the y term.

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

3. Isolate the term containing d(y)/d(x) by subtracting 3*x2 from both sides.

3*y2d(y)/d(x)=−3*x2

4. Solve for dy/dx by dividing both sides by 3*y2

d(y)/d(x)=(−3*x2)/(3*y2)

5. Simplify the fraction by canceling the common factor of 3.

d(y)/d(x)=−(x2)/(y2)

Final Answer

d(y)/d(x)=−(x2)/(y2)

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