Compute the first three derivatives of \( f(t)=\sqrt{12 t^{2}+11} \) \( f^{\prime}(t)=\square \) \( f^{\prime \prime}(t)=\square \) \( f^{\prime \prime \prime}(t)=\square \)

To compute the first three derivatives of the function \( f(t) = \sqrt{12t^2 + 11} \), we will use the chain rule and the power rule. ### First Derivative \( f'(t) \) 1. Rewrite the function: \[ f(t) = (12t^2 + 11)^{1/2} \] 2. Apply the chain rule: \[ f'(t) = \frac{1}{2}(12t^2 + 11)^{-1/2} \cdot (24t) \] 3. Simplify: \[ f'(t) = \frac{24t}{2\sqrt{12t^2 + 11}} = \frac{12t}{\sqrt{12t^2 + 11}} \] Thus, the first derivative is: \[ f'(t) = \frac{12t}{\sqrt{12t^2 + 11}} \] ### Second Derivative \( f''(t) \) To find the second derivative, we will differentiate \( f'(t) \). 1. Use the quotient rule: \[ f''(t) = \frac{(12\sqrt{12t^2 + 11}) \cdot (1) - (12t) \cdot \left(\frac{1}{2}(12t^2 + 11)^{-1/2} \cdot (24t)\right)}{(12t^2 + 11)} \] 2. Simplify the expression: \[ f''(t) = \frac{12\sqrt{12t^2 + 11} - \frac{12t \cdot 12t}{\sqrt{12t^2 + 11}}}{12t^2 + 11} \] \[ = \frac{12\sqrt{12t^2 + 11} - \frac{144t^2}{\sqrt{12t^2 + 11}}}{12t^2 + 11} \] \[ = \frac{12(12t^2 + 11) - 144t^2}{(12t^2 + 11)\sqrt{12t^2 + 11}} \] \[ = \frac{12(11 - 132t^2)}{(12t^2 + 11)\sqrt{12t^2 + 11}} \] Thus, the second derivative is: \[ f''(t) = \frac{12(11 - 12t^2)}{(12t^2 + 11)\sqrt{12t^2 + 11}} \] ### Third Derivative \( f'''(t) \) To find the third derivative, we will differentiate \( f''(t) \). 1. Use the quotient rule again: \[ f'''(t) = \frac{(12(12t^2 + 11)\sqrt{12t^2 + 11}) \cdot (0 - 144t) - (12(11 - 12t^2)) \cdot \left(\frac{1}{2}(12t^2 + 11)^{-1/2} \cdot (24t)\right)}{(12t^2 + 11)^2} \] 2. This will be quite complex, but we can simplify it step by step. The full expression will involve careful handling of the terms. After performing the calculations, we find: \[ f'''(t) = \frac{(12(12t^2 + 11)(-144t) + 12(11 - 12t^2)(12t)}{(12t^2 + 11)^{3/2}} \] Thus, the third derivative is: \[ f'''(t) = \frac{-1728t(12t^2 + 11) + 144t(11 - 12t^2)}{(12t^2 + 11)^{3/2}} \] ### Summary of Derivatives \[ f'(t) = \frac{12t}{\sqrt{12t^2 + 11}} \] \[ f''(t) = \frac{12(11 - 12t^2)}{(12t^2 + 11)\sqrt{12t^2 + 11}} \] \[ f'''(t) = \text{(complex expression as derived above)} \] The exact form of \( f'''(t) \) can be simplified further, but it