To find the derivative $f'(t)$ of the function $f(t) = 5t^4 - 6t + 2e^t$, we will differentiate each term separately.

1. First term: $5t^4$

   • Using the power rule $\frac{d}{dt}[t^n] = n t^{n-1}$, we get: $\frac{d}{dt}[5t^4] = 4 \cdot 5t^{3} = 20t^3$

2. Second term: $-6t$

   • Using the rule $\frac{d}{dt}[t] = 1$, we get: $\frac{d}{dt}[-6t] = -6$

3. Third term: $2e^t$

   • The derivative of $e^t$ is simply $e^t$, so: $\frac{d}{dt}[2e^t] = 2e^t$

Now, summing these derivatives together:

$f'(t) = 20t^3 - 6 + 2e^t$

Thus, the derivative of $f(t)$ is:

$f'(t) = 20t^3 - 6 + 2e^t$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the second derivative $f''(t)$ of $f(t) = 5t^4 - 6t + 2e^t$?
2. How would you find the critical points of $f(t) = 5t^4 - 6t + 2e^t$?
3. What is the behavior of $f(t)$ as $t \to \infty$?
4. Can you apply the chain rule to find the derivative of a composite function involving $f(t)$?
5. What are the applications of derivatives in real-world problems?

Tip: When finding derivatives, always check if each term involves a product, quotient, or composite function, as these require additional rules (product rule, quotient rule, or chain rule).