`tan^(3)x`

Answer

To differentiate \( \tan^3 x \), we will use the chain rule. Here's the step-by-step solution:

### Step 1: Identify the outer and inner functions
Let \( y = \tan^3 x \). We can identify the outer function as \( u^3 \) where \( u = \tan x \).

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( u^3 \) with respect to \( u \) is:
\[ \frac{dy}{du} = 3u^2 \]

### Step 3: Differentiate the inner function
Now, we need to differentiate the inner function \( u = \tan x \). The derivative of \( \tan x \) is:
\[ \frac{du}{dx} = \sec^2 x \]

### Step 4: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 3u^2 \cdot \sec^2 x \]

### Step 5: Substitute back for \( u \)
Now, substitute back \( u = \tan x \):
\[ \frac{dy}{dx} = 3(\tan x)^2 \cdot \sec^2 x \]

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

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