Question

Find the derivatives of the functions:$f\left(x\right)={\left(e^x\right)}^e$

Short answer

The required answer is$e^{ex+1}$.

Step-by-step solution

Step 1. Given information.  

The given function is:$f\left(x\right)={\left(e^x\right)}^e$

Step 2. Find the derivative of the given function.

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