What is the derivative of #(5x)/e^x#?

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Answer 1 · 2016-09-05T14:56:37

#dy/dx=5e^-x(1-x)#.

Let #y=5x/e^x=5x*e^-x#

#:. dy/dx=d/dx(5x*e^-x)=5d/dx(x*e^-x)#.

#:. dy/dx=5[x*d/dx(e^-x)+e^-x*d/dx(x)]#

#=5{x*e^-x*d/dx(-x)+e^-x*1].............."[Chain Rule]"#

#=5(-xe^-x+e^-x)#

#:. dy/dx=5e^-x(1-x)#.

Alternatively , using Quotient Rule ,

#dy/dx=5d/dx(x/e^x)=5{(e^xd/dx(x)-xd/dx(e^x))/(e^x)^2}#

#=5((e^x-xe^x)/(e^x)^2)=5((e^x(1-x))/(e^x)^2)=(5(1-x))/e^x#

#:. dy/dx=5e^-x(1-x)#, as before!

Enjoy Maths.!