Question

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

Short answer

The derivative \(f'(x)\) of the function \(f(x) = \ln (x+2)\) is \(f'(x) = \frac{1}{x+2}\). The domain of \(f'(x)\) is all real numbers except \(-2\).

Step-by-step solution

Differentiation

Differentiate the function using chain rule. Recall that the derivative of \(ln(u)\) is \(\frac{1}{u}\) times the derivative of \(u\). So if we let \(u = x + 2\), then \(f'(x) = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{x+2} \cdot 1\)

Computation

The derivative of \(x+2\) is 1, hence the derivative of \(f(x) = \ln(x+2)\) turns out to be \(f'(x) = \frac{1}{x+2}\)

Domain Determination

It is known that the denominator of a fraction cannot be zero, as division by zero is undefined. Thus, to find the domain of \(f'(x)\) we need to determine where the denominator \(x+2\) is not equal to zero, which occurs for all real numbers except \(-2\). Therefore, the domain of \(f'(x)\) is all \(x\) such that \(x \neq -2\)