If g (x) = x/e^x find g^ (n) (x)

This equation is a very short question that has a very long answer,

Let me show you how to get the answer,

g (x) = x/e^x

One thing you can do is find the first few derivatives, and then you can validate the solution by induction.

g' (x) = [ (1) e^x - x (e^x) ] / (e^x) ^2

= [e^x - xe^x] / [e^x]^2

= (e^x) (1 - x) / (e^x) ^2

= (1 - x) / e^x

g'' (x) = [ (-1) (e^x) - (1 - x) (e^x) ] / (e^x) ^2

= [ - e^x - e^x + x e^x ] / (e^x) ^2

= [ - 2e^x + x e^x ] / (e^x) ^2

= (e^x) (-2 + x) / (e^x) ^2

= (-2 + x) / (e^x)

= (-1) (2 - x) / e^x

g''' (x) = [ (1) (e^x) - (-2 + x) (e^x) ] / (e^x) ^2

g''' (x) = [ e^x - (-2e^x + x e^x) ] / (e^x) ^2

g''' (x) = [ e^x + 2e^x - x e^x ] / (e^x) ^2

g''' (x) = [ 3e^x - x e^x ] / (e^x) ^2

g''' (x) = (e^x) (3 - x) / (e^x) ^2

g''' (x) = (3 - x) / e^x

By inspection, it appears the pattern is

g^ (n) (x) = (-1) ^ (n - 1) (n - x) / e^x