question given the function $f(x)=e^{3x}$, determine the slope of the tangent line of $f$ at $x = - 5$ using…

question given the function $f(x)=e^{3x}$, determine the slope of the tangent line of $f$ at $x = - 5$ using the limit shown below. you do not have to simplify your answer. answer attempt 1 out of 2 $lim_{h\rightarrow0}$

question given the function $f(x)=e^{3x}$, determine the slope of the tangent line of $f$ at $x = - 5$ using the limit shown below. you do not have to simplify your answer. answer attempt 1 out of 2 $lim_{h\rightarrow0}$

Answer

Explanation:

Step1: Recall the limit - definition of the derivative

The slope of the tangent line of a function $y = f(x)$ at $x=a$ is given by $f^{\prime}(a)=\lim_{h\rightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a=-5$ and $f(x)=e^{3x}$.

Step2: Substitute $a=-5$ into the limit - definition

First, find $f(-5)=e^{3\times(-5)}=e^{-15}$ and $f(-5 + h)=e^{3(-5 + h)}=e^{-15+3h}$. Then, the limit is $\lim_{h\rightarrow0}\frac{f(-5 + h)-f(-5)}{h}=\lim_{h\rightarrow0}\frac{e^{-15 + 3h}-e^{-15}}{h}$.

Answer:

$\lim_{h\rightarrow0}\frac{e^{-15 + 3h}-e^{-15}}{h}$