Question

How do you find the derivative of `x/(e^(2x))`?

Answer 1

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

Explanation:

`"differentiate using the "color(blue)"quotient rule"`

`"Given "f(x)=(g(x))/(h(x))" then"`

`f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"`

`g(x)=xrArrg'(x)=1`

`h(x)=e^(2x)rArrh'(x)=e^(2x)xxd/dx(2x)=2e^(2x)`

`rArrd/dx(x/(e^(2x)))`

`=(e^(2x)-2xe^(2x))/(e^(2x))^2`

`=(e^(2x)(1-2x))/(e^(2x))^2=(1-2x)/(e^(2x))`

Answer 2

`(dy)/(dx)=e^(-2x)(1-2x)=(1-2x)/(e^(2x))`

Explanation:

we can arrange this function so that we can use the product rule

`y=x/(e^(2x))`

`=>y=xe^(-2x)`

the product rule

`d/(dx)(uv)=v(du)/(dx)+u(dv)/(dx)`

`(dy)/(dx)=e^(-2x)d/(dx)(x)+xd/(dx)(e^(-2x))`

`(dy)/(dx)=e^(-2x)-2xe^(-2x)=e^(-2x)(1-2x)`