Question: Differential of $e^{e^x}$ with respect to $x$ is:

Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]

To differentiate a composite function of the form $f(g(x))$, we apply the Chain Rule of differentiation:

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

We use the standard derivative formula: $\frac{d}{dx}(e^u) = e^u \cdot \frac{du}{dx}$.

Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]

Let $y = e^{e^x}$. Differentiating both sides with respect to $x$:

$$ \frac{dy}{dx} = \frac{d}{dx} (e^{e^x}) $$

Applying the chain rule, treat $e^x$ as the inner function $u$:

$$ \frac{dy}{dx} = e^{e^x} \cdot \frac{d}{dx}(e^x) $$

Since the derivative of $e^x$ is $e^x$ itself:

$$ \frac{dy}{dx} = e^{e^x} \cdot e^x $$

Step 3 (Final Concluded Answer): [0.25 Mark]

Rearranging the terms, we get:

$$ \frac{dy}{dx} = e^x e^{e^x} $$

Comparing this with the given options, the correct choice is (C).

💡 EL10 Concept Insight: Think of this like a set of Russian nesting dolls; you differentiate the outer "e-to-the-power" layer first, keeping the inside exactly as it is, and then multiply by the derivative of the inner layer you just uncovered!