Find derivative of `f(e^("tan"x))` w.r.t. x at x = 0. It is given that f(1) = 5

Answer

To find the derivative of \( f(e^{\tan x}) \) with respect to \( x \) at \( x = 0 \), we will follow these steps:

### Step 1: Define the function
Let \( y = f(e^{\tan x}) \).

### Step 2: Differentiate using the chain rule
To find \( \frac{dy}{dx} \), we apply the chain rule:
\[ \frac{dy}{dx} = f'(e^{\tan x}) \cdot \frac{d}{dx}(e^{\tan x}) \]

### Step 3: Differentiate \( e^{\tan x} \)
Next, we need to differentiate \( e^{\tan x} \):
\[ \frac{d}{dx}(e^{\tan x}) = e^{\tan x} \cdot \sec^2 x \]

### Step 4: Substitute back into the derivative
Now substituting this back into our derivative expression:
\[ \frac{dy}{dx} = f'(e^{\tan x}) \cdot e^{\tan x} \cdot \sec^2 x \]

### Step 5: Evaluate at \( x = 0 \)
Now we need to evaluate this derivative at \( x = 0 \):
1. Calculate \( e^{\tan(0)} \):
\[ \tan(0) = 0 \implies e^{\tan(0)} = e^0 = 1 \]
2. Calculate \( \sec^2(0) \):
\[ \sec(0) = 1 \implies \sec^2(0) = 1^2 = 1 \]

Putting it all together:
\[ \frac{dy}{dx}\bigg|_{x=0} = f'(e^{\tan(0)}) \cdot e^{\tan(0)} \cdot \sec^2(0) = f'(1) \cdot 1 \cdot 1 = f'(1) \]

### Step 6: Use the given information
We are given that \( f(1) = 5 \). However, we need \( f'(1) \) to complete our derivative evaluation. Since we do not have explicit information about \( f'(1) \), we cannot compute a numerical answer without additional information.

### Final Answer
Thus, the derivative of \( f(e^{\tan x}) \) with respect to \( x \) at \( x = 0 \) is \( f'(1) \).