Question

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

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

Hint:

For functions of the form \(e^{\text{something}}\), the derivative always starts with the original function itself, followed by the derivative of the 'something'.
Chain rule: differentiate from 'outside-in'.

Answer 1

The Correct Option is C

Concept:
• The derivative of \(e^x\) with respect to \(x\) is \(e^x\).
• Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).
• For a function in the form \(e^{u(x)}\), the derivative is \(e^{u(x)} \cdot \frac{du}{dx}\).

Step 1: Identify the components for the chain rule
Let \(y = e^{e^x}\).
Let the exponent be \(u\), so \(u = e^x\).
Then the original function becomes \(y = e^u\).

Step 2: Differentiate the individual components
Differentiate \(y\) with respect to \(u\):
\[ \frac{dy}{du} = \frac{d}{du}(e^u) = e^u \] Differentiate \(u\) with respect to \(x\):
\[ \frac{du}{dx} = \frac{d}{dx}(e^x) = e^x \]

Step 3: Apply the chain rule formula
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \] Substitute the results from
Step 2:
\[ \frac{dy}{dx} = e^u \cdot e^x \] Now, replace \(u\) with the original expression \(e^x\):
\[ \frac{dy}{dx} = e^{e^x} \cdot e^x \] Rearranging for standard notation:
\[ \frac{dy}{dx} = e^x e^{e^x} \] This matches option (C).