$\frac{d}{dx}e^{e^{x^{e}}}$

Explanation:

Step1: Recall chain - rule

The derivative of $y = e^{u}$ with respect to $x$ is $\frac{dy}{dx}=e^{u}\cdot\frac{du}{dx}$ by the chain - rule. Let $u = e^{e^{x^{e}}}$.

Step2: Differentiate the outer - function

The outer - function is $y = e^{u}$, and its derivative with respect to $u$ is $e^{u}$.

Step3: Differentiate the inner - function $u = e^{v}$

Let $v = e^{x^{e}}$. The derivative of $u = e^{v}$ with respect to $v$ is $e^{v}$, and by the chain - rule, $\frac{du}{dx}=e^{v}\cdot\frac{dv}{dx}$.

Step4: Differentiate the inner - function $v = e^{w}$

Let $w=x^{e}$. The derivative of $v = e^{w}$ with respect to $w$ is $e^{w}$, and by the chain - rule, $\frac{dv}{dx}=e^{w}\cdot\frac{dw}{dx}$.

Step5: Differentiate $w = x^{e}$

The derivative of $w=x^{e}$ with respect to $x$ is $ex^{e - 1}$ (using the power - rule $\frac{d}{dx}x^{n}=nx^{n - 1}$).

Step6: Combine the results

Substitute back: $w=x^{e}$, $v = e^{x^{e}}$, $u = e^{e^{x^{e}}}$. $\frac{d}{dx}e^{e^{e^{x^{e}}}}=e^{e^{e^{x^{e}}}}\cdot e^{e^{x^{e}}}\cdot e^{x^{e}}\cdot ex^{e - 1}$.

Answer:

$e^{e^{e^{x^{e}}}}\cdot e^{e^{x^{e}}}\cdot e^{x^{e}}\cdot ex^{e - 1}$