given the function $f(x)=e^{x - 5}$, determine the derivative of $f$ at $x=-1$ using the limit shown below…

given the function $f(x)=e^{x - 5}$, determine the derivative of $f$ at $x=-1$ using the limit shown below. you do not have to simplify your answer. answer attempt 1 out of 2 $lim_{x\rightarrow - 1}$

given the function $f(x)=e^{x - 5}$, determine the derivative of $f$ at $x=-1$ using the limit shown below. you do not have to simplify your answer. answer attempt 1 out of 2 $lim_{x\rightarrow - 1}$

Answer

Explanation:

Step1: Recall the limit - definition of the derivative

The derivative of a function $y = f(x)$ at $x = a$ is given by $f^{\prime}(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x - a}$. Here, $f(x)=e^{x - 5}$, $a=-1$, and $f(-1)=e^{-1 - 5}=e^{-6}$.

Step2: Substitute into the limit - formula

$f^{\prime}(-1)=\lim_{x\rightarrow - 1}\frac{e^{x - 5}-e^{-6}}{x+1}$

Answer:

$\lim_{x\rightarrow - 1}\frac{e^{x - 5}-e^{-6}}{x + 1}$