1. **State the problem:** Differentiate the function $$P = 6t^3 + 2e^t$$ with respect to $$t$$. 2. **Recall the differentiation rules:** - The derivative of $$t^n$$ with respect to $$t$$ is $$nt^{n-1}$$. - The derivative of $$e^t$$ with respect to $$t$$ is $$e^t$$. 3. **Apply the rules to each term:** - For $$6t^3$$, the derivative is $$6 \times 3t^{3-1} = 18t^2$$. - For $$2e^t$$, the derivative is $$2 \times e^t = 2e^t$$. 4. **Combine the derivatives:** $$\frac{dP}{dt} = 18t^2 + 2e^t$$ 5. **Final answer:** The derivative of the function $$P$$ with respect to $$t$$ is $$\boxed{18t^2 + 2e^t}$$