Let's find the derivative of the function $f(x) = 4e^x + e^{x^e}$.

We can differentiate each term separately:

1. The derivative of $4e^x$ with respect to $x$ is simply $4e^x$, since the derivative of $e^x$ is $e^x$.

2. For the second term, $e^{x^e}$, we use the chain rule. The chain rule states that if $g(x)$ is a function of $x$, and $f(g(x))$ is a function of $g(x)$, then:

   $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

   In this case, $g(x) = x^e$ and $f(u) = e^u$. First, differentiate $e^u$ with respect to $u$, which gives $e^u$. Then, differentiate $x^e$ with respect to $x$:

   $\frac{d}{dx} x^e = e \cdot x^{e-1}$

   Applying the chain rule, the derivative of $e^{x^e}$ is:

   $\frac{d}{dx} e^{x^e} = e^{x^e} \cdot e \cdot x^{e-1}$

So, the derivative of the function is:

$f'(x) = 4e^x + e^{x^e} \cdot e \cdot x^{e-1}$

Final Answer:

$f'(x) = 4e^x + e^{x^e} \cdot e \cdot x^{e-1}$

Would you like a more detailed explanation of any part of this? Or do you have other questions?

Related Questions:

1. How do you apply the chain rule to more complex functions?
2. What is the derivative of $x^e$?
3. Can we simplify $e^{x^e} \cdot e \cdot x^{e-1}$?
4. What are other rules of differentiation besides the chain rule?
5. How do you differentiate logarithmic functions?

Tip:

When differentiating functions with exponents like $x^e$, remember to treat the exponent as a constant and apply the power rule with respect to $x$.