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)).
