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The general strategy to replace a composed expression like diff(y, x)/y^4 is to replace only a simple part of it by an equivalent expression. In your case:

sage: y = function('y')
sage: Y = y(x)
sage: z = -3*diff(Y, x) / Y^4
sage: z.subs(diff(Y, x) == 2/5*Y^4)
-6/5

The general strategy to replace a composed expression like diff(y, x)/y^4 is to replace only a simple part of it by an equivalent expression. In your case:

sage: y = function('y')
sage: Y = y(x)
sage: z = -3*diff(Y, x) / Y^4
x)/Y^4
sage: z.subs(diff(Y, x) == 2/5*Y^4)
-6/5