Question

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

Answer 1

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