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

Question

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

Emma Aldridge · Accepted Answer

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

We are required to locate the derivative.

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

Applying the rule of quotients, which is

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

where

#u = e^x#

#v = e^(x-2) - 4#:

#= ((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)#

This can be made simpler to

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

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

Adam Adkins · Answer

undefined To differentiate $ f(x) = \frac{e^x}{e^{x-2}-4} $ using the quotient rule, you first identify $ u $ as the numerator $ e^x $ and $ v $ as the denominator $ e^{x-2}-4 $. Then, apply the quotient rule formula:

$$ f'(x) = \frac{u'v - uv'}{v^2} $$

Where:
- $ u' $ denotes the derivative of the numerator $ e^x $, which is $ e^x $.
- $ v' $ denotes the derivative of the denominator $ e^{x-2}-4 $.
- $ v^2 $ represents the square of the denominator.

Differentiate $ e^{x-2}-4 $ to get $ e^{x-2} $.

Then, apply the quotient rule formula:

$$ f'(x) = \frac{e^x(e^{x-2}-4) - e^x(e^{x-2})}{(e^{x-2}-4)^2} $$

Simplify the expression:

$$ f'(x) = \frac{e^x(e^{x-2} - 4 - e^{x-2})}{(e^{x-2}-4)^2} $$
$$ f'(x) = \frac{e^x(e^{x-2} - e^{x-2} - 4)}{(e^{x-2}-4)^2} $$
$$ f'(x) = \frac{e^x(-4)}{(e^{x-2}-4)^2} $$

This is the derivative of the function $ f(x) $.