Question

Compute y' for the equation xy¹ = y⁵?

Answer 1

To find y', differentiate the equation xy¹ = y⁵ using the power rule and isolate y'.

Step-by-step explanation:

To find y', we need to differentiate the equation with respect to x. The equation is xy¹ = y⁵. To differentiate, we will use the power rule. Let's break it down step by step:

1. Differentiate the product of x and y¹: (xy¹)' = (x)'(y¹) + (x)(y¹)'
2. Simplify: y + xy' = x(y⁵)'
3. Differentiate y⁵: (y⁵)' = 5y⁴(y)'
4. Plug in the values and solve for y': y + xy' = xy⁵'
5. Simplify further and isolate y': xy' = xy⁵' - y
6. Finally, divide both sides by x to solve for y': y' = (xy⁵' - y)/x