Questions: Let f(t) = sqrt(6) / t^5. Then, f'(t) = □ f'(2) = □

Solution Steps

To solve this problem, we need to find the derivative of the function \( f(t) = \frac{\sqrt{6}}{t^5} \) and then evaluate this derivative at \( t = 2 \).

1. Find the derivative \( f'(t) \):

   • Use the power rule for differentiation. Rewrite the function as \( f(t) = \sqrt{6} \cdot t^{-5} \).
   • Differentiate using the power rule: \( \frac{d}{dt} [t^n] = n \cdot t^{n-1} \).

2. Evaluate the derivative at \( t = 2 \):

   • Substitute \( t = 2 \) into the derivative to find \( f'(2) \).

Given the function: \[ f(t) = \frac{\sqrt{6}}{t^5} \]

To find the derivative \( f'(t) \), we rewrite the function as: \[ f(t) = \sqrt{6} \cdot t^{-5} \] Using the power rule \( \frac{d}{dt} [t^n] = n \cdot t^{n-1} \), we get: \[ f'(t) = \sqrt{6} \cdot (-5) \cdot t^{-6} = -5 \cdot \frac{\sqrt{6}}{t^6} \]

Substitute \( t = 2 \) into the derivative: \[ f'(2) = -5 \cdot \frac{\sqrt{6}}{2^6} = -5 \cdot \frac{\sqrt{6}}{64} \]

Final Answer