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

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: \[ f^{(3)}(x) = 16 \sec^4(x) \tan(x) + 8 \tan^3(x) \sec^2(x) \]