How do you differentiate # x/(e^(2x))#?

1 Answer
Jim G.
May 31, 2016

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

Explanation:

differentiate using the #color(blue)"quotient and chain rules"#

Quotient rule :

f(x) = g(x).h(x) then #f'(x)=(h(x).g'(x)-g(x).h'(x))/(h(x)^2#

Chain rule :

#d/dx(f(g(x))=f'(g(x)).g'(x)#
#"-------------------------------------------------------------"#
#g(x)=xrArrg'(x)=1#

#h(x)=e^(2x)rArrh'(x)=e^(2x).2=2e^(2x)#
#"------------------------------------------------------------"#
Substitute these values into f'(x) in the quotient rule

#f'(x)=(e^(2x).1-x.2e^(2x))/(e^(2x))^2=(e^(2x)-2xe^(2x))/e^(4x)#

#=(e^(2x)(1-2x))/e^(4x)=(1-2x)/e^(2x)#

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