質問: if g(x)=x/e^x find g^(n)(x) if g(x)=x/e^x find g^(n)(x)
回答: 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