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

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Answer 1 · 2017-07-15T02:58:13

#d/(dx) [(e^x)/(e^(x-2)-4)] = color(blue)((-4e^x)/((e^(x-2)-4)^2)#

We're asked to find the derivative

#d/(dx) [(e^x)/(e^(x-2) - 4)]#

Using the quotient rule, which is

#d/(dx) [u/v] = (v(du)/(dx) - u(dv)/(dx))/(v^2)#

where

#= ((e^(x-2)-4)(d/(dx)[e^x]) - (e^x)(d/(dx)[e^(x-2)-4]))/((e^(x-2)-4)^2)#

The derivative of #e^x# is #e^x# (fundamental!):

#= ((e^(x-2)-4)(e^x) - (e^x)(d/(dx)[e^(x-2)-4]))/((e^(x-2)-4)^2)#

The derivative of #e^(x-2)# is thus also #e^(x-2)#:

#= ((e^(x-2)-4)(e^x) - (e^x)(e^(x-2)))/((e^(x-2)-4)^2)#

#= color(blue)(((e^(x-2)-4)(e^x) - e^(2x-2))/((e^(x-2)-4)^2)#

You can simplify this to

#= (e^(2x-2)-4e^x - e^(2x-2))/((e^(x-2)-4)^2)#

#= color(blue)((-4e^x)/((e^(x-2)-4)^2)#