Question

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

Short answer

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

Step-by-step solution

Step 1. Given information.  

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

Step 2. Find the derivative of the given function.

$$\begin{gathered} f\left(x\right)=e^{x^e} \\ f\text{'}\left(x\right)=\hspace{0.17em}\frac{\partial }{\partial \hspace{0.17em}x}\left(e^{x^e}\right) \\ =e^{x^e}\frac{d}{dx}\left(x^e\right) \\ =e^{x^e}e{x}^{e-1} \\ =e^{x^e+1}{x}^{e-1} \end{gathered}$$