Question

In Exercises \(37-42,\) find \(f^{\prime}(x)\) and state the domain of \(f^{\prime}\) $$f(x)=\ln (2 x+2)$$

Short answer

The derivative of the function is \(f^{\prime}(x) = \frac{1}{x+1}\) and its domain is \(x \in (-\infty, -1) \cup (-1, +\infty)\).

Step-by-step solution

Apply the Chain Rule to Find the Derivative

The derivative of the function \(f(x)=\ln (2 x+2)\) can be calculated using the chain rule, which states that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. The derivative of a logarithmic function \(\ln u\), where \(u\) is a function of \(x\), can be written as \(\frac{1}{u}\). Therefore, the derivative of the function is \(f^{\prime}(x) = \frac{1}{2x+2}\times\frac{d(2x+2)}{dx}.\)

Calculate the Derivative of the Inside Function

The inside function is \(2x+2\). Its derivative, calculated using the power rule, is \(\frac{d(2x+2)}{dx}=2\). Substituting this in the previous equation gives \(f^{\prime}(x) = \frac{1}{2x+2}\times 2\).

Simplify the Derivative

After simplifying, the derivative of the function becomes \(f^{\prime}(x) = \frac{2}{2x+2} = \frac{1}{x+1}.\)

Find the Domain of the Derivative

The derivative is defined when the denominator is not equal to zero. Thus we set \(x+1 \neq 0\), giving \(x \neq -1\). Therefore, the domain of the derivative is \(-\infty<x<-1\) and \(-1<x<\infty\), which is the same as \(x \in (-\infty, -1) \cup (-1, +\infty)\).