1. **State the problem:** Differentiate the function $f(t) = 6e^t \left(5\sqrt{t} + \frac{1}{2t^7}\right)$ with respect to $t$. 2. **Recall the product rule:** If $f(t) = u(t)v(t)$, then $f'(t) = u'(t)v(t) + u(t)v'(t)$. 3. **Identify components:** Let $u(t) = 6e^t$ and $v(t) = 5t^{1/2} + \frac{1}{2}t^{-7}$. 4. **Differentiate $u(t)$:** $u'(t) = 6e^t$ because the derivative of $e^t$ is $e^t$. 5. **Differentiate $v(t)$:** $$v'(t) = 5 \cdot \frac{1}{2} t^{-1/2} + \frac{1}{2} \cdot (-7) t^{-8} = \frac{5}{2} t^{-1/2} - \frac{7}{2} t^{-8}$$ 6. **Apply the product rule:** $$f'(t) = u'(t)v(t) + u(t)v'(t) = 6e^t \left(5t^{1/2} + \frac{1}{2} t^{-7}\right) + 6e^t \left(\frac{5}{2} t^{-1/2} - \frac{7}{2} t^{-8}\right)$$ 7. **Combine terms inside the parentheses:** $$f'(t) = 6e^t \left(5t^{1/2} + \frac{1}{2} t^{-7} + \frac{5}{2} t^{-1/2} - \frac{7}{2} t^{-8}\right)$$ 8. **Final answer:** $$\boxed{f'(t) = 6e^t \left(5t^{1/2} + \frac{1}{2} t^{-7} + \frac{5}{2} t^{-1/2} - \frac{7}{2} t^{-8}\right)}$$