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 $$\boxed{18t^2 + 2e^t}$$