Question

given the function f(x)=4ln(x + 3), write an expression that represents the derivative of f using the limit shown below. you do not have to simplify your answer.

Explanation:

The limit - definition of the derivative of a function \(y = f(x)\) is \(f^\prime(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\).

Step1: Identify \(f(x + h)\)

Given \(f(x)=4\ln(x + 3)\), then \(f(x + h)=4\ln((x + h)+3)=4\ln(x + h+3)\)

Step2: Substitute into the limit - definition

Substitute \(f(x + h)\) and \(f(x)\) into the formula \(f^\prime(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\).
We get \(f^\prime(x)=\lim_{h
ightarrow0}\frac{4\ln(x + h+3)-4\ln(x + 3)}{h}\)

Answer:

\(\lim_{h
ightarrow0}\frac{4\ln(x + h+3)-4\ln(x + 3)}{h}\)