Question

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

Answer 1

`(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)`