Question

let (f(x)=ln(x - 3)). which is greater: the instantaneous rate of change of (f) at (x = 4) or the instantaneous rate of change of (f) at (x = 10)? use the graph of (f) to justify your answer.

Explanation:

Step1: Find the derivative of $f(x)$

The derivative of $y = \ln(u)$ is $y'=\frac{u'}{u}$. For $f(x)=\ln(x - 3)$, let $u=x - 3$, then $u'=1$. So $f'(x)=\frac{1}{x - 3}$.

Step2: Calculate the instantaneous rate - of - change at $x = 4$

Substitute $x = 4$ into $f'(x)$: $f'(4)=\frac{1}{4 - 3}=1$.

Step3: Calculate the instantaneous rate - of change at $x = 10$

Substitute $x = 10$ into $f'(x)$: $f'(10)=\frac{1}{10 - 3}=\frac{1}{7}$.

Step4: Compare the two values

Since $1>\frac{1}{7}$, the instantaneous rate of change of $f$ at $x = 4$ is greater.

Answer:

The instantaneous rate of change of $f$ at $x = 4$ is greater.