Differentiate `y = e^(tanx) ` w.r. to x

Answer

To differentiate the function \( y = e^{\tan x} \) with respect to \( x \), we can follow these steps:

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

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we use the chain rule. The chain rule states that if you have a composite function \( y = e^{u} \) where \( u = \tan x \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

### Step 3: Differentiate \( e^{u} \)
First, we differentiate \( e^{u} \):
\[ \frac{dy}{du} = e^{u} = e^{\tan x} \]

### Step 4: Differentiate \( \tan x \)
Next, we differentiate \( u = \tan x \):
\[ \frac{du}{dx} = \sec^2 x \]

### Step 5: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = e^{\tan x} \cdot \sec^2 x \]

### Step 6: Write the final answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = e^{\tan x} \sec^2 x \]