$$ f ( x ) = an ^ { 2 } ( x ) , f ^ { i i i } ( x ) $$

Question

$$ f ( x ) = 	an ^ { 2 } ( x ) , f ^ { i i i } ( x ) $$

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To find the third derivative $$ f^{(3)}(x) $$ of the function $$ f(x) = \tan^2(x) $$, we will first compute the first and second derivatives step by step.

1. **First Derivative**:
   We start with the function:
   $$
   f(x) = \tan^2(x)
   $$
   Using the chain rule, we have:
   $$
   f'(x) = 2 \tan(x) \cdot \sec^2(x)
   $$
   This can be simplified to:
   $$
   f'(x) = 2 \tan(x) \sec^2(x)
   $$

2. **Second Derivative**:
   Now we differentiate $$ f'(x) $$:
   $$
   f''(x) = \frac{d}{dx} \left( 2 \tan(x) \sec^2(x) \right)
   $$
   We will use the product rule here. Let $$ u = 2 \tan(x) $$ and $$ v = \sec^2(x) $$:
   $$
   f''(x) = u'v + uv'
   $$
   where:
   - $$ u' = 2 \sec^2(x) $$
   - $$ v' = 2 \sec^2(x) \tan(x) $$

Thus,
   $$
   f''(x) = (2 \sec^2(x)) \sec^2(x) + (2 \tan(x))(2 \sec^2(x) \tan(x))
   $$
   Simplifying this gives:
   $$
   f''(x) = 2 \sec^4(x) + 4 \tan^2(x) \sec^2(x)
   $$

3. **Third Derivative**:
   Now we differentiate $$ f''(x) $$:
   $$
   f^{(3)}(x) = \frac{d}{dx} \left( 2 \sec^4(x) + 4 \tan^2(x) \sec^2(x) \right)
   $$
   We will differentiate each term separately.

For the first term $$ 2 \sec^4(x) $$:
   $$
   \frac{d}{dx} (2 \sec^4(x)) = 8 \sec^4(x) \tan(x)
   $$

For the second term $$ 4 \tan^2(x) \sec^2(x) $$:
   Using the product rule again:
   $$
   \frac{d}{dx} (4 \tan^2(x) \sec^2(x)) = 4 \left( 2 \tan(x) \sec^2(x) \cdot \sec^2(x) + \tan^2(x) \cdot 2 \sec^2(x) \tan(x) \right)
   $$
   Simplifying this gives:
   $$
   = 8 \tan(x) \sec^4(x) + 8 \tan^3(x) \sec^2(x)
   $$

Combining both results, we have:
   $$
   f^{(3)}(x) = 8 \sec^4(x) \tan(x) + 8 \tan(x) \sec^4(x) + 8 \tan^3(x) \sec^2(x)
   $$
   This simplifies to:
   $$
   f^{(3)}(x) = 16 \sec^4(x) \tan(x) + 8 \tan^3(x) \sec^2(x)
   $$

Thus, the third derivative of $$ f(x) = \tan^2(x) $$ is:
$$
f^{(3)}(x) = 16 \sec^4(x) \tan(x) + 8 \tan^3(x) \sec^2(x)
$$
The third derivative of $$ f(x) = \tan^2(x) $$ is $$ f^{(3)}(x) = 16 \sec^4(x) \tan(x) + 8 \tan^3(x) \sec^2(x) $$.

\( f ( x ) = \tan ^ { 2 } ( x ) , f ^ { i i i } ( x ) \)

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