Differentiate `tan^(4)x`.

Options

A. `4tan^(3)x sec^(2)x`
B. `tan^(3)x sec^(2)x`.
C. `4tan^(4)x sec^(2)x`.
D. `4tan^(3)x secx`.

Explanation

To differentiate \( \tan^4 x \), we will follow these steps:

### Step 1: Identify the function
Let \( y = \tan^4 x \).

### Step 2: Apply the chain rule
To differentiate \( y \), we will use the chain rule. According to the chain rule, if \( y = u^n \) where \( u = \tan x \) and \( n = 4 \), then:
\[ \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \]
Here, \( u = \tan x \) and \( n = 4 \).

### Step 3: Differentiate the outer function
Now, applying the chain rule:
\[ \frac{dy}{dx} = 4 \cdot \tan^{3} x \cdot \frac{du}{dx} \]

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

### Step 5: Substitute back into the derivative
Now substituting \( \frac{du}{dx} \) back into the equation:
\[ \frac{dy}{dx} = 4 \cdot \tan^{3} x \cdot \sec^2 x \]

### Final Result
Thus, the derivative of \( \tan^4 x \) is:
\[ \frac{dy}{dx} = 4 \tan^{3} x \sec^2 x \]

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