derivative of `e^(tanx)`

Answer

To find the derivative of the function \( f(x) = e^{\tan x} \), we will apply the chain rule. Here are the steps to solve the problem:

### Step 1: Identify the outer and inner functions
The function can be expressed as:
- Outer function: \( e^u \) where \( u = \tan x \)
- Inner function: \( u = \tan x \)

### Step 2: Differentiate the outer function
The derivative of the outer function \( e^u \) with respect to \( u \) is:
\[ \frac{d}{du}(e^u) = e^u \]

### Step 3: Differentiate the inner function
Now, we differentiate the inner function \( u = \tan x \) with respect to \( x \):
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

### Step 4: Apply the chain rule
Using the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
\[ \frac{d}{dx}(e^{\tan x}) = e^{\tan x} \cdot \sec^2 x \]

### Step 5: Write the final answer
Thus, the derivative of \( e^{\tan x} \) is:
\[ \frac{d}{dx}(e^{\tan x}) = e^{\tan x} \sec^2 x \]

### Summary of the solution:
The derivative of \( e^{\tan x} \) is \( e^{\tan x} \sec^2 x \).

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