Chapter 4.3, Problem 3QR

To determine

To calculate : The domain of $f$ and $f^{\prime }$ for the function $f\left(x\right)=x{e}^x$ .

Answer to Problem 3QR

The domain of $f$ and $f^{\prime }$ for the function $f\left(x\right)=x{e}^x$ is $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information : The function $f\left(x\right)=x{e}^x$ .

Formula used : Use the product rule to calculate the differentiation $f^{\prime }$ .

Calculation :

Consider the function $f\left(x\right)=x{e}^x$ ,

The domain of the function is the set of real numbers except the values where the given function is undefined.

The exponential function is defined for all the values of real numbers.

So, the domain of $f\left(x\right)=x{e}^x$ is set of real numbers that is $\left(-\infty ,\infty \right)$ .

Now, to find the derivative of given function use the product rule of differentiation.

The value of $f^{\prime }$ for the function $f\left(x\right)=x{e}^x$ is calculated as follows.

  $f^{\prime }\left(x\right)=x{e}^x+e^x$

As the exponential function is defined for all the real values, so the derivative is also defined for all real values that is $\left(-\infty ,\infty \right)$ .

Thus, the domain of $f$ and $f^{\prime }$ for the function $f\left(x\right)=x{e}^x$ is $\left(-\infty ,\infty \right)$ .