Let y((n)) indicate the nth derivative:

First we will solve the given differential equation





We may swap means or extremes in any proportion:











Using the quotient rule for derivative:



Using the quotient rule for derivative again:



Using the quotient rule for derivative again:



So apparently it looks as though the nth derivative is this, and since the
signs alternate, we multiply by the sign-changing factor (-1)n: 



We can prove this inductively since we have two cases where the rule works.

Assume n=k is some case where that rule works, then



Using the quotient rule:



which is this assumed formula for n = k+1



So we have proven inductively that



Now we show that the given formula for the nth derivative is equivalent to 
the equation we proved by substituting  for y



 

Multiplying top and bottom by x

 

This is the same as the equation we proved. So the claim is true.

Edwin