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

2 Answers
Jim G.
Mar 18, 2018

(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))

sjc
Mar 18, 2018

(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)

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