Proof:

Given: y = x/x^5

To prove: x(dy/dx) = y(1 - y)

Differentiating both sides of the equation with respect to x using the quotient rule, we get:

dy/dx = (x^5 * d/dx(x) - x * d/dx(x^5)) / (x^5)^2
= (x^5 - 5x^4) / x^10
= (x^5(1 - 5/x)) / x^10
= (1 - 5/x) / x^5

Now, let's simplify the right-hand side of the equation:

y(1 - y) = (x/x^5) * (1 - x/x^5)
= (x/x^5) * (x^5/x^5 - x/x^5)
= (x/x^5) * (x^5 - x) / x^5
= (x/x^5) * (x^5(1 - 1/x)) / x^5
= (x/x^5) * (1 - 1/x)
= (1 - 1/x) / x^4

Comparing the left-hand side (x(dy/dx)) and the right-hand side (y(1 - y)), we can see that they are equal:

x(dy/dx) = (1 - 5/x) / x^5
= (1 - 1/x) / x^4
= y(1 - y)

Therefore, we have proven that x(dy/dx) = y(1 - y) for the given equation y = x/x^5.