Question

given the function $f(x)=3ln(x - 2)$, determine the instantaneous rate of change of $f$ at $x = 7$ using the limit shown below. you do not have to simplify your answer.

Explanation:

Step1: Recall the limit - definition of the derivative

The instantaneous rate of change of a function $y = f(x)$ at $x=a$ is given by $f^{\prime}(a)=\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 7$ and $f(x)=3\ln(x - 2)$. So, $f(7)=3\ln(7 - 2)=3\ln(5)$ and $f(7 + h)=3\ln((7 + h)-2)=3\ln(5 + h)$.

Step2: Substitute into the limit - formula

$f^{\prime}(7)=\lim_{h
ightarrow0}\frac{3\ln(5 + h)-3\ln(5)}{h}=3\lim_{h
ightarrow0}\frac{\ln(5 + h)-\ln(5)}{h}$.
Using the property of logarithms $\ln m-\ln n=\ln\frac{m}{n}$, we can rewrite it as $3\lim_{h
ightarrow0}\frac{\ln(\frac{5 + h}{5})}{h}=3\lim_{h
ightarrow0}\frac{\ln(1+\frac{h}{5})}{h}$.

Answer:

$3\lim_{h
ightarrow0}\frac{\ln(1+\frac{h}{5})}{h}$