Chapter 4.4, Problem 37E

(a)

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 37E

The derivative is $\frac{dy}{dx}=\frac{1}{x+2}$ .

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $f\left(x\right)=\ln \left(x+2\right)$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}y=f\left(x\right)\\ \frac{dy}{dx}=\frac{d\left(\ln \left(x+2\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{1}{x+2}\times \frac{d\left(x+2\right)}{dx}\\ \frac{dy}{dx}=\frac{1}{x+2}\end{array}$

So the derivative of is $\frac{dy}{dx}=\frac{1}{x+2}$ .

(b)

To determine

To calculate: The domain of the differentiated function.

Answer to Problem 37E

The domain is $\left\{x\in R:x\ne -2\right\}$ .

Explanation of Solution

Given information: The equation is given as,

  $f\left(x\right)=\ln \left(x+2\right)$

The domain is where the function is eligible to be solved,

So the domain is $\left\{x\in R:x\ne -2\right\}$ .