1. **State the problem:** Differentiate the function $$F(t) = \frac{At^2}{Bt^5 + Ct^9}$$ with respect to $t$. 2. **Recall the quotient rule:** For a function $$\frac{u(t)}{v(t)}$$, the derivative is $$\frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}$$. 3. **Identify parts:** - $$u(t) = At^2$$ - $$v(t) = Bt^5 + Ct^9$$ 4. **Compute derivatives:** - $$u'(t) = 2At$$ - $$v'(t) = 5Bt^4 + 9Ct^8$$ 5. **Apply quotient rule:** $$F'(t) = \frac{(2At)(Bt^5 + Ct^9) - (At^2)(5Bt^4 + 9Ct^8)}{(Bt^5 + Ct^9)^2}$$ 6. **Expand numerator:** $$= \frac{2ABt^{6} + 2ACt^{10} - 5ABt^{6} - 9ACt^{10}}{(Bt^5 + Ct^9)^2}$$ 7. **Combine like terms:** $$= \frac{(2ABt^{6} - 5ABt^{6}) + (2ACt^{10} - 9ACt^{10})}{(Bt^5 + Ct^9)^2} = \frac{-3ABt^{6} - 7ACt^{10}}{(Bt^5 + Ct^9)^2}$$ 8. **Final answer:** $$F'(t) = \frac{-3ABt^{6} - 7ACt^{10}}{(Bt^5 + Ct^9)^2}$$