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

##### 1 Answer

May 22, 2017

# f'(x) = (2e^x(1 - x))/(e^(x-2) +2x)^2 #

We apply the Quotient Rule for Differentiation:

# d/dx(u/v) = (v(du)/dx-u(dv)/dx)/v^2 # , or less formally,# " "(u/v)' = (v(du)-u(dv))/v^2 #

I was taught to remember the rule in word; " *vdu minus udv all over v squared* ". To help with the ordering I was taught to remember the acronym, VDU as in Visual Display Unit.

So with

# { ("Let, "u=e^x, => , (du)/dx=e^xx), ("And, "v=e^(x-2) +2x, =>, (dv)/dx=e^(x-2) +2 ) :}#

Then:

# f'(x) = ((e^(x-2) +2)(e^x) - (e^(x-2) +2x)(e^x))/(e^(x-2) +2x)^2 #

# " " = e^x*(e^(x-2) +2 - e^(x-2) - 2x)/(e^(x-2) +2x)^2 #

# " " = e^x*(2 - 2x)/(e^(x-2) +2x)^2 #

# " " = (2e^x(1 - x))/(e^(x-2) +2x)^2 #