Question

The differential of \(e^{e^x}\) with respect to \(x\) is:

• A. \(\log x\)
• B. \(e^{e^x}\)
• C. \(e^x \cdot e^{e^x}\)
• D. \((e^x)^2\)

Hint:

When applying the chain rule to layered exponential equations like \(e^{f(x)}\), the original term \(e^{f(x)}\) always remains entirely intact in the final product, multiplied directly by the derivative of its exponent, \(f'(x)\).

Answer 1

The Correct Option is C

Concept: To find the derivative of a composite function of the form \(y = f(g(x))\), we must utilize the standard Chain Rule of differential calculus. The chain rule states: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] For exponential terms, we use the fundamental derivative rule: \[ \frac{d}{dx}(e^u) = e^u \cdot \frac{du}{dx} \]

Step 1: Set up the differentiation problem
Let the given function be defined explicitly as: \[ y = e^{e^x} \] Here, the outer function can be envisioned as \(f(u) = e^u\) where the core nested function variable is defined as \(u = g(x) = e^x\).

Step 2: Differentiate using the Chain Rule
Differentiating \(y\) with respect to \(x\) requires treating the exponent \(e^x\) as our inner function: \[ \frac{dy}{dx} = \frac{d}{dx}\left(e^{e^x}\right) = e^{e^x} \cdot \frac{d}{dx}(e^x) \] We know that the derivative of the natural exponential function \(e^x\) with respect to \(x\) is simply itself: \[ \frac{d}{dx}(e^x) = e^x \]

Step 3: Combine terms to form final solution
Substitute this component back into our multi-part differentiation expression: \[ \frac{dy}{dx} = e^{e^x} \cdot e^x \] Rearranging the commutative components cleanly: \[ \frac{dy}{dx} = e^x \cdot e^{e^x} \] This perfectly aligns with the given choice option (C).