Find dy/dx x^4+y^4=24xy

Problem

x^4+y^4=24*x*y

Solution

1. Differentiate both sides with respect to x treating y as a function of x and applying the chain rule to the y^4 term.

d(x^4)/d(x)+d(y^4)/d(x)=(d(24)*x*y)/d(x)

2. Apply the power rule to the left side and the product rule to the right side.

4*x^3+4*y^3d(y)/d(x)=24*y+24*xd(y)/d(x)

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

4*y^3d(y)/d(x)−24*xd(y)/d(x)=24*y−4*x^3

4. Factor out the common factor d(y)/d(x) from the left side.

d(y)/d(x)*(4*y^3−24*x)=24*y−4*x^3

5. Solve for the derivative by dividing both sides by (4*y^3−24*x)

d(y)/d(x)=(24*y−4*x^3)/(4*y^3−24*x)

6. Simplify the fraction by dividing the numerator and the denominator by the common factor 4

d(y)/d(x)=(6*y−x^3)/(y^3−6*x)

Final Answer

d(y)/d(x)=(6*y−x^3)/(y^3−6*x)

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