`tan^(2)x `

Explanation

To find the differentiation of \( \tan^2 x \), we can follow these steps:

### Step 1: Define the function
Let
\[ y = \tan^2 x \]

### 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 \( u = \tan x \), so we can rewrite \( y \) as:
\[ y = u^2 \]

### Step 3: Differentiate \( y \) with respect to \( u \)
Now, differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 2u \]

### Step 4: Differentiate \( u \) with respect to \( x \)
Next, we need to find \( \frac{du}{dx} \):
\[ \frac{du}{dx} = \frac{d}{dx}(\tan x) = \sec^2 x \]

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

### Step 6: Substitute back for \( u \)
Substituting back \( u = \tan x \):
\[ \frac{dy}{dx} = 2 \tan x \cdot \sec^2 x \]

### Final Answer
Thus, the derivative of \( \tan^2 x \) is:
\[ \frac{dy}{dx} = 2 \tan x \sec^2 x \]

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