Find dy/dx y^4=y^2-x^2

Problem

y4=y2−x2

Solution

1. Differentiate both sides with respect to x using the chain rule for terms involving y

d(y4)/d(x)=d(y2−x2)/d(x)

2. Apply the power rule and chain rule to the left side and the first term on the right side.

4*y3d(y)/d(x)=2*yd(y)/d(x)−2*x

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

4*y3d(y)/d(x)−2*yd(y)/d(x)=−2*x

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

d(y)/d(x)*(4*y3−2*y)=−2*x

5. Isolate the derivative by dividing both sides by (4*y3−2*y)

d(y)/d(x)=(−2*x)/(4*y3−2*y)

6. Simplify the fraction by dividing the numerator and the denominator by 2

d(y)/d(x)=(−x)/(2*y3−y)

Final Answer

d(y)/d(x)=(−x)/(2*y3−y)

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