We have: #f(x) = - frac(e^(x))(x - 2)#
First, let's find the function for the slope, #f'(x)#, by differentiating #f(x)#:
#Rightarrow f'(x) = frac(d)(dx)(- frac(e^(x))(x - 2))#
#Rightarrow f'(x) = - frac(d)(dx)(frac(e^(x))(x - 2))#
Using the quotient law of differentiation:
#Rightarrow f'(x) = - frac((x - 2) cdot frac(d)(dx)(e^(x)) - (e^(x)) cdot frac(d)(dx)(x - 2))((x - 2)^(2))#
#Rightarrow f'(x) = - frac((x - 2) cdot e^(x) - (e^(x)) cdot 1)((x - 2)^(2))#
#Rightarrow f'(x) = - frac(x e^(x) - 2 e^(x) - e^(x))((x - 2)^(2))#
#Rightarrow f'(x) = - frac(x e^(x) - 3 e^(x))((x - 2)^(2))#
#therefore f'(x) = - frac(e^(x) (x - 3))((x - 2)^(2))#
Now, we need to evaluate the slope at #x = - 2#.
So let's substitute #-2# in place of #x#:
#Rightarrow f'(- 2) = - frac(e^((- 2)) ((- 2) - 3))(((- 2) - 2)^(2))#
#Rightarrow f'(- 2) = - frac(e^(- 2) (- 5))((- 4)^(2))#
#Rightarrow f'(- 2) = - frac(- frac(5)(e^(2)))(16)#
#therefore f'(- 2) = frac(5)(16 e^(2))#
Therefore, at #x = - 2#, the slope of #f(x)# is #frac(5)(16 e^(2))#.
