How do you differentiate f(x) = (e^x-4x)/(2x-e^x) using the quotient rule?

1 Answer
Jan 23, 2017

The answer is =(2e^x(1-x))/(2x-e^x)^2

Explanation:

The quotient rule is

(u/v)'=(u'v-uv')/(v^2)

Here,

u=e^x-4x, =>, u'=e^x-4

v=2x-e^x, =>, v'=2-e^x

Therefore,

f'(x)=((e^x-4)(2x-e^x)-(e^x-4x)(2-e^x))/(2x-e^x)^2

=(2xe^x-cancele^(2x)-cancel8x+4e^x-2e^x+cancele^(2x)+cancel8x-4xe^x)/(2x-e^x)^2

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

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