To find the derivative of the given function, let's break it down step by step.

The function is:

$f(t) = \frac{6}{5} t e^t$

Step 1: Identify the type of function.

This is a product of two functions:

• $g(t) = \frac{6}{5} t$
• $h(t) = e^t$

We'll need to apply the Product Rule for differentiation, which states that:

$\frac{d}{dt}[g(t) h(t)] = g'(t) h(t) + g(t) h'(t)$

Step 2: Differentiate each function.

• The derivative of $g(t) = \frac{6}{5} t$ is:

$g'(t) = \frac{6}{5}$

• The derivative of $h(t) = e^t$ is:

$h'(t) = e^t$

Step 3: Apply the Product Rule.

Using the product rule:

$f'(t) = g'(t) h(t) + g(t) h'(t)$

Substituting the values:

$f'(t) = \frac{6}{5} e^t + \frac{6}{5} t e^t$

Final answer:

The derivative of the function is:

$f'(t) = \frac{6}{5} e^t (1 + t)$

Let me know if you'd like more details or further clarifications!

Here are 5 related questions to explore:

1. How would the derivative change if the exponential was $e^{2t}$ instead of $e^t$?
2. What happens if the constant $\frac{6}{5}$ is replaced by a variable? How would the derivative look?
3. Can you apply the product rule to more complex functions, such as products of three functions?
4. What are the applications of this derivative in physics or engineering problems?
5. How would you compute the second derivative of this function?

Tip: When differentiating products of functions, the product rule is always essential. Remember to carefully differentiate each function involved before applying the rule.