Differentiate the following w.r.t.x. `e^(e^(x))`

Answer

To differentiate the function \( y = e^{e^x} \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the function
Let:
\[ y = e^{e^x} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

In our case:
- Let \( f(u) = e^u \) where \( u = e^x \).
- Then, \( y = f(u) \).

### Step 3: Differentiate the outer function
First, we differentiate the outer function \( f(u) = e^u \):
\[ f'(u) = e^u \]

### Step 4: Differentiate the inner function
Next, we differentiate the inner function \( u = e^x \):
\[ \frac{du}{dx} = e^x \]

### Step 5: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = f'(u) \cdot \frac{du}{dx} = e^{e^x} \cdot e^x \]

### Step 6: Write the final answer
Thus, the derivative of \( y = e^{e^x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = e^{e^x} \cdot e^x \]

### Summary
The final answer is:
\[ \frac{dy}{dx} = e^{e^x} \cdot e^x \]

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