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Answer 1 · 2017-03-02T09:03:16

#d^n/(dx)^n(1/x) = (-1)^n(n!)/x^(n+1)#

We have that:

#(1) f^((n))(x) = (-1)^n(n!)x^(-(n+1))#

We can prove it directly for #n=1#:

#f'(x) = d/dx(1/x) = -1/x^2 = (-1)^1(1!)x^(-(1+1))#

In general, suppose:

#f^((n))(x) = (-1)^n(n!)x^(-(n+1))#

we have that:

#f^((n+1))(x) = d/dx f^((n))(x) = (-1)^n(n!)(-(n+1))x^(-(n+2)) = (-1)^(n+1)((n+1)!)x^(-(n+2))#

So the equation (1) is proven by induction.