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.

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

Short answer

The required function is$f\left(x\right)=\log (1+e^x)$

Step-by-step solution

Step 1. Given Information  

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

Step 2. Calculation 

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

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

Thus, we get the required function.