Question

In Exercises 59–63, find a function f that has the given derivative f'. In each case you can find the answer with an educated guess-and-check process.

$f\text{'}\left(x\right)=e^x(1+e^x)$

Short answer

The required function is $f\left(x\right)=\left(e^x+\frac{e^{2x}}{2}\right)$

Step-by-step solution

Step 1. Given Information  

The given derivative is$f\text{'}\left(x\right)=e^x(1+e^x)$

Step 2. Calculation  

We will use a targeted guess and check method to find an antiderivative of given derivative.

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}\left(e^x+\frac{e^{2x}}{2}\right) \\ =e^x+\frac{e^{2x}}{2}·2 \\ =e^x+e^{2x} \\ =e^x(1+e^x) \end{gathered}$$

Thus, we get the required function.