1. **Problem statement:** Given the function $F(t) = -50t e^{-0.2t} - 250 e^{-0.2t}$, find the function $f(t)$ and its first derivative $f'(t)$. 2. **Rewrite the function:** Notice that both terms share the factor $e^{-0.2t}$, so we can factor it out: $$F(t) = e^{-0.2t}(-50t - 250)$$ 3. **Define $f(t)$:** We set $$f(t) = -50t - 250$$ 4. **Find the first derivative $f'(t)$:** Differentiate $f(t)$ with respect to $t$: $$f'(t) = \frac{d}{dt}(-50t - 250) = -50$$ 5. **Summary:** - The function is $f(t) = -50t - 250$ - Its first derivative is $f'(t) = -50$ This completes the solution.